These lecture notes are an extended form of a course given at a CIMPA master class held in LIMA, Peru, in the summer of 2008. The students who attended these lectures were already introduced to linear differential equations, Gevrey asymptotics, k-summability and resurgence by my colleagues Michèle Loday, Claude Mitschi and David Sauzin. The aim was merely to show the resurgent methods acting on an example and along that line, to extend the presentation of the resurgence theory of Jean Ecalle provided that the need.

The present lecture notes reflect this plan and this pedagogical point of view. The example that we follow along this course is the First Painlevé differential equation, or Painlevé I for short. Besides its simplicity, various reasons justify this choice. One of them is the non-linearity, which is the field where the resurgence theory reveals its power. Another reason lies on the fact that resonances occur, a case which is scarcely found in the literature. Last but not least, the Painlevé equations and their transcendents appear today to be an inescapable knowledge in analysis for young mathematicians. It was thus certainly worthy to detail the complete resurgent structure for Painlevé I.

We have tried to be as self-contained as possible. Nevertheless, the reader is assumed to have a previous acquaintance with the theories of summability, especially with Borel-Laplace summation and a little background with resurgence theory. Since this volume deals with ordinary non-linear differential equations, we begin with definitions and phenomena linked to the non-linearity. Special attention is then brought to Painlevé I and to its so-called tritruncated and truncated solutions that are constructed by proving the summability of the transseries solutions. We analyze the formal integral for Painlevé I and, equivalently, the formal transform that brings Painlevé I to its normal form. We detail the resurgent structure for Painlevé I via additional material in resurgence theory. As a rule, each chapter ends with some comments on possible extensions for which we provide references to the existing literature.

• f is analytic on U, that is f can be represented by a convergent power series in a neighbourdhood of each x ∈ U; • f is complex differentiable on U;

• f is weakly holomorphic, that is f is continuous on U and partially differentiable on U with respect to each variable

x i (x = (x 1 , • • • , x n )).
As a matter of fact, it is enough to assume only the holomorphy in each complex variable without the continuity hypothesis (Hartogs theorem).

Theorem 1.1 (Cauchy problem). Let U ⊂ C × C m be a domain and F : U → C m be a holomorphic vector function. For every (x 0 , u 0 ) ∈ U, there exist a polydisc D(x 0 , ε 0 ) ∏ 1≤i≤m D(u 0i , ε i ) ⊂ U and a solution u : D(x 0 , ε 0 ) → ∏ 1≤i≤m D(u 0i , ε i ) of the analytic ODE of order 1 and dimension m, du dx = F(x, u), which satisfies the initial value condition u(x 0 ) = u 0 . Moreover this solution is unique, u belongs to O(D(x 0 , ε 0 )) and also depends holomorphically on the initial value u 0 .

In what follows we shall consider essentially scalar ODEs, that it ODEs of dimension 1 and of order N. The theorem 1.1 translates to this case as well, since every ODE of order N and of dimension 1, once solved in his highest derivative, is equivalent to an ODE of order 1 and of dimension N : if u = v 0 , u = v 1 , • • • , u (N-1) = v N-1 , the following Cauchy problem,

u (N) = F(x, u, • • • , u (N-1) ) u(x 0 ), • • • , u (N-1) (x 0 ) = u 0 , • • • , u (N-1) 0 ,
is equivalent to that one:

d dx      v 0 . . . v N-2 v N-1      =      v 1 . . . v N-1 F(x, v 0 , • • • , v N-1 )      and    v 0 . . . v N-1    (x 0 ) =    u 0 . . . u (N-1) 0    .

Some usual terminologies

The following terminologies are commonly used (see, e.g. [6]):

• The general solution of an ODE of order N and of dimension 1 is the set of all solutions determined in application of the Cauchy theorem 1.1. It depends on N arbitrary complex constants. • A particular or special solution is a solution derived from the general solution when fixing a particular initial data. • A singular solution is a solution which is not particular.

Algebraic differential equations

In a moment we shall concentrate on algebraic differential equations, these we define now.

Let U ⊂ C be a domain. We denote by M (U) the field of meromorphic functions on U. The ODE (1.1) of order N and of dimension 1 is said to be algebraic on a domain U if F ∈ M (U)[u, u , • • • , u (N) ] that is, F is polynomial in (u, u , • • • , u (N) ) with meromorphic coefficients in x. An algebraic ODE is rational if it is of degree one in the highest derivative u (N) , and linear (homogeneous) if F is a linear form in (u, u , • • • , u (N) ).

On singularities of solutions of ordinary differential equations

Notations

We fix some notations that will be used in a moment.

O → C ϕ x ∈ O x → x ∈ C
which associates to a germ its support [12,9], the (non-connected) topological space O becomes an étalé space, that is q is a local homeomorphism. The analytic continuation of the germ u ∈ O x 1 along λ , if exists, is the image of the unique path Λ : [a, b] → O such that Λ (a) = u and whose projection by q is λ ,

O Λ q [a, b] -→ C λ
. With this notation, cont λ u = Λ (b). See [32] for more details.

Problem

We consider an ODE of order N and dimension 1, F x, u(x), u (x), • • • , u (N) (x) = 0 with F : U → C a holomorphic function on the open domain U ⊂ C × C N+1 . Assume that x 0 , u 0 , • • • , u

(N) 0 ∈ U and that F x 0 , u 0 , • • • , u (N) 0 = 0 ∂ N+2 F x 0 , u 0 , • • • , u (N) 0 = 0
. By the implicit function theorem, the Cauchy problem

F x, u(x), u (x), • • • , u (N) (x) = 0 u(x 0 ), • • • , u (N) (x 0 ) = u 0 , • • • , u (N) 0
is locally equivalent to a Cauchy problem where the ODE is solved in its highest derivative. Theorem 1.1 thus provides a holomorphic solution u near x = x 0 . We consider a path γ : [a, b] → C from x 0 to x 1 in C and for s ∈ [a, b] we denote by γ s : [a, s] → C the restriction to [a, s] of γ. Assume that u can be analytically continued along the path γ and that for every s ∈ [a, b], the value at γ(s) of the analytic continuation cont γ s x, u, u , • • • , u (N) along γ s belongs to U. Then the analytic continuation cont γ u along γ of the solution u still satisfies the differential equation, thanks to the uniqueness of the analytic continuation. This property raises the question of describing the singularities of the analytic continuations of solutions of analytic ODEs, for instance for an algebraic differential equation defined on an open domain. As we shall see, appearance of singularities is quite different whether one considers linear or nonlinear ODEs.

Linear differential equations

Linear differential equations are studied in [32,30], see also, e.g. [39,25,19,22]. For linear (homogeneous) ordinary differential equations it results from the Cauchy existence theorem and the Grönwall lemma that the general solution has no other singularities than the so-called fixed singularities which arise from the coefficients of the ODE once solved for the highest derivative.

Example 1

We start with an equation where x = 0 is an irregular singular point of Poincaré rank 1,

x 2 u + u = 0, u(x) = Ce 1/x , C ∈ C.
Here x = 0 is a fixed essential singularity for the general solution (but not for the particular solution u(x) = 0), which arises from the equation itself.

If u ∈ O(D (0, r)) is holomorphic on the punctured disc D (0, r) = D(0, r) \ {0}, then u can be represented by its Laurent series expansion ∑ n∈Z a n x n which converges in 0 < |x| < r.

One says that 0 is an essential singularity if and only if the Laurent series expansion has an infinite number of n < 0 such that a n = 0 or, equivalently, if u has no limit (finite or infinite) when x → 0. A typical example is provided by the function e1/x .

Example 2

We consider the Airy equation,

u -xu = 0, u(x) = C 1 Ai(x) +C 2 Bi(x), C 1 ,C 2 ∈ C.
Here Ai and Bi are the Airy's special functions of the first and second kind respectively. These are entire functions. When considered on the Riemann sphere C (see [32]), x = ∞ appears as a fixed (essential) singularity for the general solution (except again for the particular solution u(x) = 0) which arises from the equation : x = ∞ is an irregular singular point of Poincaré rank 3/2. More generally, for a linear ordinary differential equation

N ∑ k=0 a k (x)u (k) = 0, a k (x) ∈ O(U), (1.3)
the general solution can be analytically continued as a multivalued function on U \ S, S = {the zeros of a N }, or more precisely as a single valued holomorphic function once it is considered on a Riemann surface [12,9] defined as a covering space, R π ↓ U \ S

. In other words, the general solution is uniformisable (or also stable) [6] in the following sense : for any Cauchy data at x 0 ∈ U \S that determined a unique local solution u of (1.3) on a domain U 0 ⊂ U \ S, one can find a domain U 0 ⊂ R such that π| U 0 : U 0 → U 0 is a homeomorphism, and a holomorphic function φ : R → C so that φ | U 0 = u • π| U 0 .

Then, for any domain U ⊂ R so that π| U : U → U is a homeomorphism, the function φ • (π| U ) -1 is still a holomorphic solution of (1.3) on U .

Nonlinear differential equations

When nonlinear ODEs are concerned, beside the possibly fixed singularities arising from the equation, the general solution has as a rule other singularities which depend on the arbitrary coefficients : these are movable singularities.

Example 1

We consider the following nonlinear ODE, xuu 2 = 0, general solution :

u(x) = 1 C -log(x) , C ∈ C.
singular solution : u(x) = 0

For the general solution, x = 0 is a fixed branch point singularity which comes from the equation. The general solution u is uniformisable : considered as a function on the Riemann surface ( C, π) of the logarithm, C = {x = re iθ | r > 0, θ ∈ R},

π : x ∈ C → • x= re iθ ∈ C \ {0}
, one sees that the general solution u is meromorphic with poles at π -1 (e C ) : these are movable singularities, depending on the chosen coefficient C.

Example 2

The above example is just a special case of a more general rational ODE of order 1, the Riccati equation, u = a 0 (x) + a 1 (x)u + a 2 (x)u 2 a i ∈ M (U), (1.4) where U ⊂ C is a domain. By the change of unknown function u = -1 a 2 (x)

d dx log v,
equation (1.4) is linearizable into the following linear ODE,

v + a 2 (x) a 2 (x) -a 1 (x) v + a 2 (x)a 0 (x)v = 0.
The general solution for this linear equation has (fixed) singularities located at the poles of a 2 (x) a 2 (x) a 1 (x) and a 2 (x)a 0 (x). We denote by S ⊂ U this set of poles. The general solution of the Riccati equation (1.4) is then uniformisable since it can be analytically continued as a meromorphic function on a Riemann surface defined as a covering over U \ S.

When the a i belong to O(U), then the general solution of (1.4) is a meromorphic function on U [26].

Example 3

Another well known equation is the following algebraic nonlinear ODE of order 1, of degree 2 in its highest derivative, namely the elliptic equation: A particular solution is provided by the Weierstrass p-function ℘(x; g 2 , g 3 ) which can be obtained as the inverse function of the elliptic integral of the first kind

u 2 = 4u 3 -g 2 u -g 3 , (g 2 , g 3 ) ∈ C. (1.5)
x = u ∞ dq 4q 3 -g 2 q -g 3 , dx du 2 = 1 4u 3 -g 2 u -g 3 .
(Just apply the inverse function theorem).

When the discriminant D = g 3 2 -27g 2 3 satisfies the condition D = 0, the polynomial function 4u 3 g 2 ug 3 = 4(ue 1 )(ue 2 )(ue 3 ) has 3 distinct simple roots e 1 , e 2 , e 3 . In that case the elliptic function ℘(x; g 2 , g 3 ) is a doubly periodic meromorphic function with double poles at the period lattice

mω 1 + nω 2 , (n, m) ∈ Z 2 , ω 1 ω 2 / ∈ R.
The period lattice can be described as follows : consider the elliptic curve L = {(q, p) ∈ C 2 , p 2 = 4q 3g 2 qg 3 } for D = 0. The homology group H 1 (L ; Z) is a free Z-module of rank 2 and we denote by γ 1 and γ 2 two cycles which generate H 1 (L ; Z). Then the period lattice is generated by the period integrals ω 1 = e 1 dq 4q 3 g 2 qg 3 ).

The homology group H 1 (L ; Z) can be seen as a local system on C 2 \ N(D) (that is a locally constant sheaf of Z-modules on C 2 \ N(D)), where N(D) is the zero set of D.

Viewed as functions of (g 2 , g 3 ), ω 1,2 can be analytically continued as "multivalued" analytic functions on C 2 \ N(D). On the discriminant locus N(D), the solutions degenerate into simply periodic solutions, with a string of poles instead of a double array.

Conversely, starting from the period lattice with ω 1 ω 2 / ∈ R, the Weierstrass ℘function can be obtained by a series,

℘(x; g 2 , g 3 ) = x -2 + ∑ ω =0 {(x -ω) -2 -ω -2 } = x -2 + g 2 x 2 20 + g 3 x 4 28 + • • •
where the first summation extends over all ω = mω 1 + nω 2 = 0, (n, m) ∈ Z 2 while

g 2 = 60 ∑ ω =0 ω -4 , g 3 = 140 ∑ ω =0 ω -6 .
The general solution of (1.5) is given by ℘(xx 0 ; g 2 , g 3 ), since (1.5) is an autonomous ODE.

To go further on the nice properties of elliptic functions see, e.g. [37].

Example 4

The singularities of differential equations may be isolated singularities such as poles, branch points of finite or infinite determinations, or essential singularities. They may be also essential singular lines, or even perfect sets of singular points. For instance, the general solution of the following Chazy equation of class III, u (3) -2uu (2) + 3u 2 = 0, (1.6) is defined only inside or outside an open disc whose boundary is a natural movable boundary determined by the initial data [3,4].

The Painlevé program, Painlevé property and Painlevé equations

At the end of the 19th century a list of special transcendental functions was known, most of them being obtained as solutions of linear algebraic differential equations.

An algebraic function u in one complex variable x is a solution of a polynomial equation P(x, u) = 0, P ∈ C[x, u]. A transcendental function u is a function which is not algebraic.

A challenging problem in analysis was thus to discover new transcendental functions defined by algebraic ODEs which cannot be expressed in term of solutions of linear algebraic ODEs : these new functions should thus be defined by non-linear algebraic differential equations [6,8,22].

For that purpose a systematic approach needs first to classify the ODEs under convenient criters. This is the goal of the so-called Painlevé program (see [6] and references therein) which consists in classifying all algebraic ODEs of first order, then second order, etc ..., whose general solution can be analytically continued as a single valued function 1 . In other words, no branch point is allowed. For instance the elliptic equation (1.5) or the Chazy equation (1.6) are such equations.

According to what we have seen, the Painlevé program splits into two problems:

• absence of fixed branch point for the general solution;

• absence of movable branch point for the general solution : this condition is the so-called Painlevé property.

In the literature, the term "Painlevé property" is sometimes used for the stronger property for the general solution of an ODE to be meromorphic, see [6] Notice that the Painlevé property for an algebraic ODE F (x, u, u , • • • , u (p) ) = 0 defined on a domain U ⊂ C is preserved by:

• a holomorphic change of variable x ∈ U → X = h(x), h ∈ O(U); • a linear fractional change of the unknown with coefficient holomorphic in U (action of the homographic group),

u → v = a(x)u + b(x) c(x)u + d(x) , v → u = d(x)v -b(x) -c(x)v + a(x) ,
a, b, c, d ∈ O(U), adbc = 0. Therefore, the classification in the Painlevé program is made up to these transformations.

1 Some elements about ordinary differential equations Notice however that other actions preserving the Painlevé property can be considered, see [6,7,22].

ODEs of order one

We consider (nonlinear) ODEs of the form

F x, u, u = 0, (1.7)
with F ∈ M (U) [u, u ]. For that class of ODEs, the Painlevé program can be considered as being achieved and we mainly refer to [20,18,6,22] for the classification.

In that case no essential movable singular point can appear ( [20], Sect. 13.6). Therefore looking for ODEs of type (1.7) with the Painlevé property reduces in asking that the movable singular points are just poles.

When (1.7) is a rational ODE, then the class of ODEs we are looking for is represented only by the Riccati equation (1.4). See [26], in particular the Malmquist-Yosida-Steinmetz type theorems.

The ODEs of type (1.7) of degree ≥ 2 in the highest derivative and satisfying the Painlevé property essentially reduce (up to the transformations mentioned above) to the elliptic equation (1.5). See [6,20] for more precise statements.

ODEs of order two and Painlevé equations

In contrast to what happens for algebraic ODEs of order one, essential movable singular points may exist when the order is ≥ 2, making the analysis more difficult. Nevertheless, the classification is known for at least algebraic equations of order two F x, u, u , u = 0,

F ∈ M (U)[u, u , u ] (1.8)
which are rational, that is of degree one in u . Such equations enjoying the Painlevé property reduce (up to transformation) to:

• equations which can be integrated by quadrature,

• or linear equations,

• or one of six ODEs known as the Painlevé equations, the first 3 being:

(P I ) u = 6u 2 + x (P II ) u = 2u 3 + xu + α (P III ) u = u 2 u - u x + αu 2 + β x + γu 3 + δ u (1.9)
For the complete list see, e.g. [20,18,6,22]. In (1.9), α, β , γ, δ are arbitrary complex constants. Each Painlevé equation can be derived from the "master equation" P V I by some limit processes [22].

Painlevé equations have beautiful properties, see e.g. [5,22,16]. One of them is the following one: Theorem 1.2. The general solution of the Painlevé equation P J , J = I, • • • ,V I admits no singular points except poles outside the set of fixed singularities.

Therefore, the Painlevé equations have the Painlevé property but moreover, the general solution is free of movable essential singularities. Notice that the Painlevé equation should be seen as defined on the Riemann sphere C. The set of fixed singular points S J of P J is a subset of {0, 1, ∞}. For instance S I and S II are just {∞}, while S III = {0, ∞}. Theorem 1.2 thus translates as follows : the general solution of P J can be analytically continued as a meromorphic function on the universal covering of C \ S J .

Theorem 1.2 can be proved in various ways. An efficient one uses the relationship between Painlevé equations and monodromy-preserving deformation of some Fuchsian differential equations [24,23,31,22,11].

The general (global) solutions of the Painlevé equations are called the Painlevé transcendents. This refers to the fact that, for generic values of the integration constants and of the parameters of the equations, these solutions cannot be written with elementary or classical transcendental functions, a question which has been completely solved only recently with the development of the modern nonlinear differential Galois theory (see [38] and references therein. For an introduction to differential Galois theory, see [32]).

Painlevé equations and related topics

The renewed interest in Painlevé equations mainly came from theoretical physics in the seventies, with the study of PDEs of the soliton type (Boussinesq equation, Korteweg-de Vries KdV and modified Korteweg-de Vries equation mKdV, etc..): when linearized by inverse scattering transform [1], these PDEs give rise to ODEs with the Painlevé property. For instance, the Boussinesq equation u ttu xx -6(u 2 ) xx + u xxxx = 0 has a self-similar solution of the form u(x,t) = w(xt) where w is either an elliptic function or satisfies the first Painlevé equation. In the same lines, the (m)KdV hierarchy introduced by Lax in [29] (and already in substance in [28] after the work of Gardner et al [13] on the KdV equation), will later give rise to various Painlevé hierarchies which are thought of as higher-order Painlevé equations and much studied since. For instance, the first Painlevé hierarchy is of the form

(P (n) I ) d [n+1] [u] + 4x = 0, n = 1, 2, • • • (1.10)
where d [n] [u] are differential polynomials recursively determined as follows (see [36] and references therein):

d [0] [u] = 1 ∂ d [n+1] [u] = ∂ 3 -8u∂ -4u d [n] [u], ∂ = d dx , n ∈ N.
(1.11) (The first Painlevé equation is (P

I ).) See also [33] and references therein, for an asymptotic study of the Jimbo-Miwa [23] and Flaschka-Newell [10] second Painlevé hierarchies [15].

For the first and second Painlevé hierarchies, one conjectures that the solutions of each equation are meromorphic, thus satisfy the Painlevé property, but there is no proof up to our knowledge [27].

Discrete (analogues of) Painlevé equations are today the matter of an intensive research, after the pioneering work of Bessis et al [2] on the study of partition functions in 2D quantum gravity, yielding what is now known as the first discrete Painlevé equation (dP I ) when the quartic matrix model is considered:

(dP I ) w n w n+1 + w n + w n-1 = an + b + cw n , a, b, c ∈ C.

(1.12)

The first discrete Painlevé equation naturally arises in the context of orthogonal polynomials. Consider the inner product

( f | g) = +∞ -∞
f (x)g(x)w(x)dx with the exponential weight w(x) = e -NV (x) , V (x) = µ 2 x 2 + λ 4 x 4 , and look for an orthogonal polynomial sequence (p n ) n≥0 , each p n being a monic polynomial of degree n. It can be shown that the polynomials p n are governed by a three-term recurrence equation of the form xp n (x) = p n+1 (x) + r n p n-1 (x) p 0 (x) = 1, p 1 (x) = x (1. 13) where r n = h n h n-1 with (p n | p m ) = h n δ nm (δ nm is the Kronecker index). This motivates the calculations of the coefficients r n which themselves satisfy a recurrence relation of the form λ r n r n+1 + r n + r n-1 + µr n = n N (1. 14) and we recognize (dP I ). Among remarkable properties, (1.14) has a continuum limit to the first Painlevé equation when the double-scaling limit n, N → ∞, n N → t is considered. See for instance [21,14,17] and references therein.

Non commutative extensions of integrable systems have recently attracted the attention of the specialists, with non commutative (analogues of) Painlevé equations and their hierarchies as main examples, see e.g. [35]. Finally, we could hardly leave untold the important group-theoretic interpretation of Painlevé equations in the line of the work of Okamoto [34], see for instance [8] and references therein.

It is not our aim to say more about Painlevé equations in general except for the first Painlevé equation which is used in this course as field of experiments in asymptotic and resurgent analysis, and which is the matter for the next chapter. Chapter 2

The first Painlevé equation Abstract This chapter aims at introducing the reader to properties of the first Painlevé equation and its general solution. The definition of the first Painlevé equation is recalled (Sect. 2.1). We precise how the Painlevé property translates for the first Painlevé equation (Sect. 2.2), a proof of which being postponed to an appendix. We explain how the first Painlevé equation also arises as a condition of isomonodromic deformations for a linear ODE (Sect. 2.3 and Sect. 2.4). Some symmetry properties are mentioned (Sect. 2.5). We spend some times to describe the asymptotic behaviour at infinity of the solutions of the first Painlevé equation and, in particular, we introduce the truncated solutions (Sect. 2.6). We eventually briefly comment the importance of the first Painlevé transcendents for models in physics (Sect. 2.7).

The first Painlevé equation

We concentrate on the first Painlevé equation,

(P I ) u = 6u 2 + x. (2.1)
We notice that for every x 0 ∈ C and every (u 0 , u 0 ) ∈ C 2 , theorem 1.1 ensures the existence of a unique solution of (2.1), holomorphic near x 0 , satisfying the initial data u(x 0 ), u (x 0 ) = (u 0 , u 0 ).

Painlevé property for the first Painlevé equation

As already mentioned, the first Painlevé equation satisfies the Painlevé property. The following more precise result holds.

Theorem 2.1. Every solution of the Painlevé equation P I can be analytically continued as a meromorphic function on C with only double poles.

This theorem will be shown in appendix. We add the following result for completeness: Theorem 2.2. Every solution of (2.1) is a transcendental meromorphic function on C with infinitely many poles.

Proof. We just give an idea of the proof. It is easy to see that every solution u of the first Painlevé equation (2.1) is a transcendental function. Otherwise, since u is meromorphic with double poles, u should be a rational function, u(x) = P(x) Q(x) 2 . Reasoning on the degrees of P and Q, one shows that this is impossible. So every solution u is a transcendental meromorphic function. It can be then derived from the Clunie lemma in Nevanlinna theory of meromorphic functions that necessarily u has an infinite set of poles [29,14].

The above properties were well-known since Painlevé [START_REF] Painlevé | Mémoire sur les équations différentielles dont l'intégrale générale est uniforme[END_REF]. The following one was also known by Painlevé, however its complete proof has been given only recently [38], see also [5].

Theorem 2.3. A solution of P I cannot be described as any combination of solutions of first order algebraic differential equations and those of linear differential equations on C.

First Painlevé equation and isomonodromic deformations condition

Each Painlevé equation P J is equivalent to a nonautonomous Hamiltonian system [39]. Concerning the first Painlevé equation this Hamiltonian system is given by the following first Painlevé system:

(H I )          du dx = ∂ H I ∂ µ = µ dµ dx = - ∂ H I ∂ u = 6u 2 + x , H I (u, µ, x) = 1 2 µ 2 -2u 3 -xu. (2.
2)

It is known [12,40] that this Hamiltonian system arises as a condition of isomonodromic deformations of the following (Schlesinger type) second order linear ODE,

(S L I )          ∂ 2 Ψ ∂ z 2 = Q I (z; u, µ, x)Ψ Q I (z; u, µ, x) = 4z 3 + 2xz + 2H I (u, µ, x) - µ z -u + 3 4(z -u) 2 .
(2. 3) In other words, u is solution of the first Painlevé equation (2.1) if and only if the monodromy data of (2.3) do not depend on x. We explain this point. Equation (2.3) has two fixed singularities z = u, ∞, so that any (local) solution of (2.3) can be analytically continued to a Riemann surface which covers C \ {u, ∞}. The point z = u is a regular singular point, and a local analysis easily shows that the monodromy at this point (see [37]) of any fundamental system of solutions of (2.3) does not depend on x. The other singular point z = ∞ is an irregular singular point. Thus the only nontrivial monodromy data of (2.3) are given by the Stokes coefficients at z = ∞.

The second order linear ODE (2.3) is equivalent to a first order linear ODE in dimension two. Each Stokes matrix is a two by two unipotent matrix (see [31,37]), and thus depends on a single complex coefficient called a Stokes coefficient.

In general these Stokes coefficients depend on x, except when Ψ satisfies the following isomonodromic deformation condition:

(D I ) ∂Ψ ∂ x = A I ∂Ψ ∂ z - 1 2
∂ A I ∂ z Ψ , A I = 1 2(z -u) (2.4)
The first Painlevé system (2.2) ensures the compatibility between equations (2.3) and (2.4) : solving a Painlevé equation is thus equivalent to solving an inverse monodromy problem (Riemann-Hilbert problem) [37,18,17,25,26,42,24,21,11].

We add another property : we mentioned that the asymptotics of (2.3) at z = ∞ are governed by some Stokes coefficients s i = s i (u, µ, x). It can be shown that the space of Stokes coefficients makes a complex manifold M I of dimension 2. Also, for any point of M I there exists a unique solution of the first Painlevé equation (2.1) for which the monodromy data of equation (2.3) are equal to the corresponding coordinates of this point [25].

Lax formalism

There is another fruitful alternative to get the Painlevé equations, however related to the previous one, based on the linear representations of integrable systems through the Lax formalism [30]. We exemplify this theory for Painlevé I, for which the socalled Lax pair A and B are the matrix operators given as follows [17]:

A = v(x) 4 z -u(x) z 2 + u(x)z + u(x) 2 + x/2 -v(x)
, B = 0 2 z/2 + u(x) 0 .

To the matrix operator A one associates a first order ODE in the z variable, whose time evolution (the x variable) is governed by another first order ODE determined by the matrix operator B,

     ∂Ψ ∂ z = AΨ ∂Ψ ∂ x = BΨ (2.5)
The compatibility condition . From what have been previously seen, the zero curvature condition allows to think of (2.5) as an isomonodromic deformations condition for its first equation.

∂ 2 Ψ ∂ z∂ x = ∂ 2 Ψ ∂ x∂

Symmetries

We would like to notice here that the cyclic symmetry group of order five acts on the set of solutions (2.1). Indeed, introducing ω k = e 2iπ 5 k , k = 0, • • • , 4, then any solution u of (2.1) is mapped to another solution u k through the transformation

u k (x) = ω 2 k u ω k x , k = 0, • • • , 4.
In general u and u k will be different solutions, an obvious exception being when u satisfies the initial data u(0) = u (0) = 0.

Asymptotics at infinity

Our aim in this section is to describe all the possible behaviors at infinity of the solutions of the first Painlevé equation (2.1). We first notice that x = ∞ is indeed a fixed singularity for P I : making the change of variable u(x) = u(t), t = 1 x , equation (2.1) translates into t 5 u + 2t 4 u = 1 + 6tu 2 , where t = 0 appears as a (irregular) singular point.

We mention that, when analysing the asymptotics of solutions of differential equations at singular points, there is a great difference between linear and nonlinear ODEs. When a linear ODE is concerned, the asymptotics of every solution can be derived from the asymptotics of a fundamental system of solutions. For non linear ODEs some care has to be taken, since as a rule singular solutions may exist, which cannot be deduced from the general solution.

The study of all possible behaviors at infinity was first made by Boutroux [3,4]. Various approaches can be used: a direct asymptotic approach in the line of Boutroux as in [15,19,22], or another one based on the relationship between the first Painlevé equation and a convenient Schlesinger type linear ODE as described in Sect. 2.3, see [25] (see also [26,28,27,42] for an exact semiclassical variant).

Dominant balance principle

We only want to give a rough idea of how to get the whole possible asymptotic behaviors and, in the spirit of this course, we follow the viewpoint of asymptotic as in [15,22,19]. In this approach, for a given ODE, the first task is to determine the terms in the equation which are dominant and of comparable size when x → ∞ along a path or a inside a sector. The reduced equation obtained by keeping the dominant terms only in the ODE gives the leading behavior. One usual trick to guess the asymptotics of solutions of ODEs is the dominant balance principle [2]. A maximal dominant balance corresponds to the case where there is a maximal set of dominant terms of comparable size in the equation. As a rule, this gives rise to the general behavior. The remaining cases are called subdominant balances.

It is useful to introduce the following notations:

• f g when x → ∞ along a path if lim x→∞ f (x) g(x) = Cte, Cte ∈ C . • f g when x → ∞ along a path if lim x→∞ f (x) g(x) = 0.
The unique maximal balance for P I consists in assuming all the three terms in (2.1) of comparable size when x → ∞. In particular u 2 and x have comparable size, so that u(x) = x 1 2 O(1) when x → ∞. We therefore write u(x) = x 1 2 v z(x) with z(x) → ∞ and v z(x) = O(1) when x → ∞. If z(x) behaves like a fractional power of x at infinity, then z (x) z(x) z (x) z (x)

1
x and this is what will be assumed. We also make the following remark : if v z is an analytic function whose asymptotics at infinity is governed by a (formal, possibly Laurent) series, then v (z)

z 2 v (z) z v(z) at infinity, that is v z(x) z 2 (x) v z(x) z(x) v z(x) when x → ∞.
Here we will adjust the choice of z(x) by adding the demand:

v z(x) z(x)v z(x) z(x) 2 v z(x) when x → ∞.
These assumptions on v and z(x) provide the identities:

u (x) = x -1 2 z(x)v z(x) O(1) + o(1), u (x) = x -3 2 z 2 (x)v z(x) O(1) + o(1)
.

Thus, if v z(x) = v z(x) = v z(x) = O(1)
and demanding that u and x have comparable size, one gets z(x) = x 5 4 O(1) as a necessary condition. This suggests with Boutroux [3,4] to make the following transformation,

u(x) = αx 1 2 v(z), z = β x 5 4 , (2.6)
with α, β = 0 some constants, under which equation (2.1) becomes:

v + v z - 4 25 v z 2 - 96α 25β 2 v 2 - 16 25αβ 2 = 0.
one finally gets:

v = 1 2 v 2 - 1 2 - v z + 4 25 v z 2 . (2.8)
We now concentrate on this equation (2.8) and we examine the possible balances.

Maximal balance, elliptic function-type behavior

We consider the maximal balance case, that is we assume that v and its derivatives can be compared to unity. This means that equation (2.8) is asymptotic to the equa-

tion v = 1 2 v 2 - 1 2
whose solutions1 are the functions v(z) = 12℘(zz 0 ; 1 12 , g 3 ) where ℘ is the Weierstrass p-function (cf. Sect. 1.2.4), while z 0 and g 3 are two free complex parameters. This indeed provides the general behaviour of the Painlevé transcendents near infinity [3,4,22] : for |z| large enough in each open quadrants the generic solution v of (2.8) has an approximate period lattice of poles, Fig. 2.1.

Q k = {z ∈ C, k π 2 < arg z < (k + 1) π 2 }, k = 0,
In this domain, excluding small neighbourdhoods of poles, the asymptotics of such a generic solution v of (2.8) is governed by Weierstrassian elliptic functions. With Kruskal & Joshi [22] one refers to this behavior as an elliptic function-type behavior. Through the transformation (2.6)-(2.7), this translates for the Painlevé I transcendents into an asymptotic regime on the sectors:

S k = {x ∈ C, -π +k 2π 5 < arg x < -π +(k +1) 2π 5 }, k = 0, 1, 2, 3, 4 mod 5. (2.9)
When z approaches the real axis (resp. the imaginary axis), |z| large enough and in a small angular strip of width O (log |z|)/|z| , then the solution v displays a near oscillatory-type behaviour with no poles, and v(z) → -1 (resp. v(z) → +1) when |z| → ∞, see [22]. The five special rays arg x = -π + k 2π 5 , k = 0, • • • , 4 thus play an important role in the asymptotics of the solutions of Painlevé I, the general solutions having lines of poles asymptotic to these rays.

Submaximal dominant balances, truncated solutions

We now consider submaximal dominant balances, that is when v or one of its derivatives differ from order unity. As shown in [22], the single consistent case occurs when v 1 and v 1. This implies that equation (2.8) is now asymptotic to the

equation 1 2 v 2 - 1 2 = 0, that is v(z) = ±1 + o(1)
. Examining this case leads to the following result: • five complex parameter families of solutions u, the so-called intégrales tronquées (truncated solutions) after Boutroux, such that u is free of poles in two adjacent sectors S k and S k+1 for |x| large enough, and u(x) = -x 6

1 2 1 + O(x - 5
2 ) at infinity in these sectors (for a convenient determination of the square root).

• among these truncated solutions, five special solutions, each of them being free of poles in four adjacent sectors S k , S k+1 , S k+2 , S k+3 for |x| large enough, with the above asymptotics at infinity in these sectors. These are the so-called intégrales tri-tronquées (tritruncated solutions).

This theorem has various proofs, see for instance [20,35,36] for "nonconventional" approaches. We will see in this course how the resurgent analysis can be used to show theorem 2.4.

There are analogues of truncated solutions for each member (P (n) (1.10), with asymptotics at infinity of the form [8]:

I ), n = 1, 2, • • • in the first Painlevé hierarchy
u(x) = (-1) n x c n 1 n+1 1 + O(x -2n+3 2n+2 ) , c n = 2 3n+1 Γ (n + 3/2) Γ (n + 2)Γ (1/2) . (2.10)
Similar results occur for the first discrete Painlevé equation (1.12), see [23].

First Painlevé equation and physical models

As already said (Sect. 1.3.3), the Painlevé equations in general and the first Painlevé equation in particular, appear by similarity reductions of integrable PDEs. They play a significant role in others physical models, see e.g. [24] and references therein for the first Painlevé equation. This includes the description of asymptotic regime in transition layers and caustic-type domain. We exemplify this fact with the focusing nonlinear Schrödinger equation

iεΨ t + ε 2 2 Ψ xx + |Ψ | 2 Ψ = 0 (fNLS).
It is shown in [10] that when considering the (so-called) dispersionless limit ε → 0, the solutions (of convenient Cauchy problems) of (fNLS) are asymptotically governed by a tritruncated solution of the first Painlevé equation. In the same work, theoretical and numerical evidences led the authors to conjecture that the tritruncated solutions of the first Painlevé equation have the following property, shown in [7] under the naming "the Dubrovin conjecture":

ρ = max u(x 0 ) 1/2 , u (x 0 ) 2 1/3 , u 2 (x 0 )+ x 0 6 1/4 , u(x 0 )u (x 0 ) 2 + 1 24 1/5
. (2.11)

Proof. If u(x) = ∞ ∑ k=0 c k (x -x 0 ) k ∈ C{x -x 0 } solves (2.1), then      c 0 = u(x 0 ), c 1 = u (x 0 ) c 2 = 3c 2 0 + x 0 2 , c 3 = 2c 0 c 1 + 1 6 (k + 1)(k + 2)c k+2 = 6 ∑ k m=0 c m c k-m , k ≥ 2 (2.12)
Let be ρ > 0 given by (2.11) so that for any integer l ∈ [0, 3],

|c l | ≤ (l + 1)ρ l+2 . (2.13)
Assume that (2.13) is satisfied for every 0 ≤ l ≤ k + 1 for a given k ≥ 2. Then by (2.12),

(k + 1)(k + 2)|c k+2 | ≤ 6 k ∑ m=0 (m + 1)(k -m + 1)ρ k+4 ≤ (k + 1)(k + 2)(k + 3)ρ k+4 . The coefficients k ∑ m=0 (m + 1)(k -m + 1) are those of the taylor expansions of (1 -x) -4 at the origin. Indeed, for |x| < 1, 1 1 -x = ∑ k≥0 x k so that 1 (1 -x) 2 = ∑ k≥0 (k + 1)x k . Therefore 1 (1 -x) 2 2 = ∑ k≥0 k ∑ m=0 (m + 1)(k -m + 1) x k .
We conclude that (2.13) is satisfied for every l ≥ 0 and this implies that R ≥ 1 ρ , where R is the radius of convergence of the series expansion u.

Lemma 2.2. In a neighbourhood of any given point x ∈ C, there exists a oneparameter family of meromorphic solutions u of (2.1) having a pole at x. Necessarily x is a double pole and u is given by the Laurent-series expansions

u(x) = 1 (x -x) 2 - x 10 (x -x) 2 - 1 6 (x -x) 3 + c 4 (x -x) 4 + ∑ k≥6 c k (x -x) k
where c 4 ∈ C is a free parameter.

Proof. We are looking for a Laurent-series u

(x) = ∞ ∑ k=p c k (x -x) k ∈ C{x -x} 1 x -x satisfying (2.1). Necessarily p ≥ -2, c -2 = 1 or 0, c -1 = 0. Therefore, either x is a regular point, or otherwise u(x) = 1 (x -x) 2 - x 10 (x -x) 2 - 1 6 (x -x) 3 + c 4 (x -x) 4 + ∑ k≥6 c k (x -x) k
where c 4 ∈ C is a free parameter, while for k ≥ 6 the coefficients are polynomial functions of ( x, α). Indeed, one has

(k -2)(k + 5)c k+2 = 6 ∑ k m=0 c m c k-m , k ≥ 2.
We can define ρ > 0 (depending on ( x, α)) such that, for 0 ≤ l ≤ 5,

|c l | ≤ 1 3 (l + 1)ρ l+2 . (2.14)
Assume that this property is satisfied for every c l , 0 ≤ l ≤ k + 1, for a given k ≥ 4. Then

(k -2)(k + 5)|c k+2 | ≤ 2 3 k ∑ m=0 (m + 1)(k -m + 1)ρ k+4 ≤ 1 9 (k + 1)(k + 2)(k + 3)ρ k+4
and we conclude that |c k+2 | ≤ 1 3 (k + 3)ρ k+4 . Therefore (2.14) is true for every l ≥ 0 and the Laurent series converges in the punctured dics D( x, 1/ρ) .

The following notations will now be used:

• D x 0 ⊂ C is an open disc, Ω is a discrete subset of D x 0 and x 0 ∈ D x 0 \ Ω .
• u is a solution of (2.1) defines by some initial data at

x 0 ∈ D x 0 \ Ω and u is meromorphic in D x 0 \ Ω . • λ (a, b) : [0, 1] → D x 0 \ Ω denotes a C ∞ -smooth path in D x 0 \ Ω with endpoints λ (a, b)(0) = a and λ (a, b)(1) = b. When b ∈ ∂ D x 0 it is assumed that λ (a, b) is a path where b is removed (that is one considers the restriction to [0, 1[ of λ (a, b)).
Moreover we assume that the length of any subsegment λ (c, d) of λ (a, b) is less that 2|c -d|.

We mention that we use the same notation λ (a, b) for the path and its image.

• x ∈ ∂ D x 0 is a singular point for u. Lemma 2.3. Assume that u(x) = 4 ∑ k=-2 a k (x -x) k + O(|x -x| 5 ) when x → x along λ (x 0 , x)
, with a -2 = 0. Then u is meromorphic at x and u is uniquely determined by ( x, a 4 ).

Proof. Since u is solution of (2.1) which is analytic at each point of the smooth path

λ (x 0 , x), one has u (x) = 6u 2 (x)+x = 6 ∑ 4 k=-2 a k (x -x) k + O(|x -x| 5 )
2 +x when

x → x along λ (x 0 , x). This implies that the asymptotic expansion is differentiable.

This is a consequence of the mean value theorem, u(x) = u(x 0 ) +

x x 0 u (s) ds along λ (x 0 , x)
which is C ∞ -smooth, and the uniqueness of the asymptotic expansion.

With the same calculus made in the proof of lemma 2.2, we show that

u(x) = 1 (x -x) 2 - x 10 (x -x) 2 - 1 6 (x -x) 3 + a 4 (x -x) 4 + O(|x -x| 5 ).
We denote by v the meromorphic solution of (2.1) obtained in lemma 2.2 with c 4 = a 4 . We set

w(x) = v(x) -(x -x) -2 = O(|x -x| 2 ) f (x) = u(x) -v(x) = O(|x -x| 5 )
and we want to show that f = 0. We have f -12

(x-x) 2 f = g with g = 12w f + 6 f 2 , g = O(|x -x| 7
). Integrating this linear ODE yields:

f (x) = C 1 (x -x) -3 +C 2 (x -x) 4 -7(x -x) -3 x x (s -x) 4 g(s) ds + (x -x) 4 7 x x (s -x) -3 g(s) ds. Since f (x) = O(|x -x| 5 ), f is solution of the fixed-point problem f = N( f ) with N( f )(x) = -7(x -x) -3 x x (s -x) 4 g(s) ds + (x -x) 4 7 x x
(sx) -3 g(s) ds.

For x 1 ∈ λ (x 0 , x) we consider the normed vector space (B, . ),

B = { f ∈ C 0 (λ (x 1 , x)), f = O(|x -x| 5 }, f = sup x∈λ (x 1 , x) |(x -x) -5 f (x)|.
We show later that (B, . ) is a Banach space (lemma 2.4). Now for x 1 close enough from x (see lemma 2.4):

• the mapping N send the unit ball B of B into itself,

• the mapping N : B → B is contractive.

Therefore the fixed-point problem f = N( f ) has a unique solution in B by the contraction mapping theorem. Obviously this solution is f = 0 and therefore u = v.

Lemma 2.4. With notations of the proof of lemma 2.3: (B, . ) is a Banach space and the mapping N : B → B is contractive.

Proof.

(B, . ) is a Banach space. Assume that ( f p ) is a Cauchy sequence in (B, . ), ∀ε, ∃p 0 : ∀p, q > p 0 , ∀x ∈ λ (x 1 , x), |(x -x) -5 ( f p (x) -f q (x))| < ε . (2.15)
Writing g p (x) = (xx) -5 f p (x), condition (2.15) implies that for every x ∈ λ (x 1 , x) the sequence (g p (x)) is a Cauchy sequence, hence g p (x) → g(x) in C. Now making q → +∞ in (2.15) one sees that g p → g uniformaly. Therefore g ∈ C 0 (λ (x 1 , x)) and is bounded on λ (x 1 , x). Thus g = (xx) -5 f with f ∈ B.

The mapping N is contractive for x 1 close enough from x. We set

N 1 ( f )(x) = -7(x -x) -3 x x (s -x) 4 g(s) ds, N 2 ( f )(x) = (x-x) 4 7 x x (s -x) -3 g(s) ds so that N( f ) = N 1 ( f ) + N 1 ( f ). One can assume that |s -x| ≤ |x -x| for s ∈ λ (x, x)).
Also, there exist r > 0 and a > 0 such that |w(x)| ≤ a|x -x| 2 when |x -x| ≤ r. We now assume that |x 1 -x| ≤ r. For any f 1 , f 2 ∈ B and x ∈ λ (x 1 , x):

(x -x) -5 N 1 ( f 1 ) -N 2 ( f 2 ) ≤ -7(x -x) -8 x x (s -x) 4 12w(s)( f 1 (s) -f 2 (s)) + 6( f 2 1 (s) -f 2 2 (s)) ds , thus (x -x) -5 N 1 ( f 1 ) -N 2 ( f 2 ) ≤ 7|x -x| -8 12a|x -x| 11 f 1 -f 2 + 6|x -x| 14 f 1 -f 2 f 1 + f 2 Length(λ (x, x)) ≤ 14|x -x| 4 12a + 12|x -x| 3 f 1 -f 2 .
The other term of (xx) -5 N 2 ( f 1 ) -N 2 ( f 2 ) is worked out in a similar way.

Choosing x 1 close enough from x, one obtains the existence of a constant Cte ∈]0, 1[ such that for any

f 1 , f 2 ∈ B, N( f 1 ) -N( f 2 ) ≤ Cte f 1 -f 2 .
Lemma 2.5. When x → x along λ (x 0 , x) with x ∈ ∂ D x 0 a singular point for u:

1. |u(x)| + |u (x)| → +∞,
(u ) 2 = 4u 3 + 2xu -2 x x 0 u(s) ds +C (2.16)
where C ∈ C is a constant. Therefore if u is bounded x → x along λ (x 0 , x) then u is bounded as well, which contradicts the first property.

Lemma 2.6. When x → x along λ (x 0 , x), with x ∈ ∂ D x 0 a singular point for u, then:

u -3 (x) x x 0 u(s) ds → 0, |u(x)| → +∞, |u (x)| → +∞.
Proof. By lemma 2.5, we know that u is unbounded when x → x along λ (x 0 , x), so that lim sup

x→ x |u(x)| = +∞, lim inf x→ x |u -1 (x)| = 0. Reminder: lim sup x→ x f (x) = lim ε→0 sup f (x) | x ∈ λ (x 0 , x) ∩ D( x, ε) , lim inf x→ x f (x) = lim ε→0 inf f (x) | x ∈ λ (x 0 , x) ∩ D( x, ε) . Since u -3 (x) x x 0 u(s) ds ≤ |u -3 (x)|. max λ (x 0 ,x) |u|. Length(λ (x 0 , x)) for x ∈ λ (x 0 , x), it turns out that lim inf x→ x u -3 (x) x x 0 u(s) ds ≤ lim inf x→ x |u -3 (x)|. max λ (x 0 ,x)
|u|. Length(λ (x 0 , x)) .

The right hand side term vanishes because u is unbounded when x → x, thus lim inf

x→ x u -3 (x) x x 0 u(s) ds = 0.
(2.17)

In particular, for every γ > 0, for every D( x, ε), there exists x ∈ λ (x 0 , x) ∩ D( x, ε) so that u -3 (x)

x x 0 u(s) ds ≤ γ.
We make the following Assumption : u -3 (x)

x x 0 u(s) ds → 0 is a false premise.
This assumption translates into the condition : there exists γ > 0 such that, for every

D( x, ε), there exists x ∈ λ (x 0 , x) ∩ D( x, ε) so that u -3 (x) x x 0 u(s) ds ≥ γ.
By continuity, we see that for any γ > 0 small enough, there exists a sequence

x n → x, x n ∈ λ (x 0 , x), such that x n x 0 u(s) ds = γ u 3 (x n ) .
(2.18)

The arguments used in the proof of lemma 2.5 show that lim sup

n |u(x n )| = +∞.
This means that there exists a subsequence We are going to prove in several steps that the above assumption leads to a contradiction.

(x n k ) of (x n ) such that |u(x n k )| → +∞.

First step

We consider the solution h n of the Cauchy problem

   (h ) 2 = 4h 3 + 2x n h + C n with C n = C -2 x n x 0 u(s) ds h(0) = u(x n ), h (0) = u (x n ) (2.
     |h n (0)| = (2γ) -1/3 C n 1/3 1 + o(1)
|h n (0)| = 2γ -1 e iφ n + 1 1/2 C n 1/2 1 + o(1) , φ n ∈ R. (2.21) Writing h n (t) = C 1/3 n H n (X), X = C 1/6 n t, (2.22)
the function H n is solution of the following elliptic differential equation (see (1.5)) with a given initial data:

             (H ) 2 = 4H 3 + 2θ n H + 1, with θ n = x n C -2/3 n H n (0) = C -1/3 n u(x n ), |H n (0)| = (2γ) -1/3 1 + o(1) , H n (0) = C -1/2 n u (x n ), |H n (0)| = 2γ -1 e iφ n + 1 1/2 1 + o(1) .
(2.23)

From the properties of elliptic functions, H n can be analytically continued as a doubly periodic meromorphic function with double poles at the period lattice

a n + mω 1 (θ n ) + nω 2 (θ n ), (n, m) ∈ Z 2 , for some a n ∈ C and ω 1,2 (θ n ) = Cte 1,2 + O(θ n ).
Second step Next we consider the function U n satisfying to the condition:

u(x) = C 1/3 n U n (X), X = C 1/6 n (x -x n ). (2.24) From (2.1), U n is solution of the ODE U = 6U 2 + θ n + ε n X, with ε n = C -5/6 n , (2.25)
and, more precisely from (2.16):

   (U ) 2 = 4U 3 + 2θ n U + 1 + 2ε n XU - X 0 U(S) dS U n (0) = C -1/3 n u(x n ), U n (0) = C -1/2 n u (x n ) (2.26)
Third step We want to show that U n and H n are locally holomorphically equivalent: we look for a function G n holomorphic near 0 such that

U n = H n • G n with G n (X) = X + g n (X), g n (0) = 0, g n (0) = 0. (2.27)
We know from (2.23) that H n = 6H 2 n + θ n , hence from (2.25) we deduce that

2g n H n • G n + (g n ) 2 H n • G n + g n H n • G n = ε n X or else: 2g n (H n • G n ) + g n H n • G n = ε n X + (g n ) 2 H n • G n .
Multiplying both parts of this equality by H n • G n and integrating, one gets:

         w n = (H n • G n ) -2 X 0 H n • G n (S) ε n S + w 2 n (S).H n • G n (S) dS = N(w n ) g n (X) = X 0 w n (S) dS, w n (0) = 0, G n (X) = X + g n (X).
(2.28) Let D(0,

|ε n | -1/4
2 ) be the disc centred at 0 of diameter |ε n | -1/4 . We denote by

D(0, |ε n | -1/4 2 ) the disc D(0, |ε n | -1/4
2 ) deprived from the discs of diameter d(γ) around the poles and the zeros of H n . We consider a path λ (0, X 0 ) in D(0,

|ε n | -1/4 2
). In (2.28), the integrals X 0 are considered along λ (0, X) ⊂ λ (0, X 0 ). We can assume that the length of any subsegment λ (0, X) of λ (0, X 0 ) is less that 2|X|. Let be a ∈]1/4, 1/2[ and (B, . ) be the Banach space

B = { f ∈ C 0 (λ (0, X 0 ))}, f = sup x∈λ (0,X 0 ) | f (x)|. Let B be the ball B = { f ∈ B, f ≤ |ε n | a }. If w ∈ B and g(X) = X 0 w(S) dS, g ≤ sup X∈λ (0,X 0 ) X 0 w(S) dS ≤ w .Length(λ (0, X 0 )) ≤ |ε n | a-1/4 .
One can assume that d(γ) ≥ 3|ε n | a-1/4 so that

N(w) ≤ |ε n |Cte 1 (γ)|ε n | -1/2 +Cte 2 (γ)|ε n | 2a-1/4 .
Therefore N(w) ≤ |ε n | a for |ε n | small enough. Quite similarly, for w 1 , w 2 ∈ B,

N(w 1 ) -N(w 2 ) = O(|ε n | a-1/4 ) w 1 -w 2 .
We conclude by the contraction mapping theorem: N has a unique fixed point in B, for |ε n | small enough.

Final step

We have seen that for |ε n | small enough and a ∈]1/4, 1/2[,

U n (X) = H n X + g n (X) , |g(X)| ≤ |ε n | a-1/4 , X ∈ D(0, |ε n | -1/4 2 ).
Therefore,

sup X∈ D(0, |εn| -1/4 2 ) C -1/3 n u(x n + C -1/6 n X) -H n (X) = O(|ε n | a-1/4 ). (2.29) Remember that | C n | → +∞ and |ε n | = | C -5/6 n | → 0 when x n → x. If X ∈ D(0, |ε n | -1/4 2 ), then C -1/6 n X belongs to a disc of radius | C n | 1/24 deprived of some discs of radius d(γ)| C n | -1/6 . Consequently, for n large enough, ∀ x ∈ D x 0 , ∃ X ∈ D(0, |ε n | -1/4 2 ), x -(x n + C -1/6 n X) ≤ d(γ) 2 | C n | -1/6 .
Choosing x = x 0 , we see from (2.29) that u is unbounded near x 0 which is a regular point for u: contradiction.

Therefore, u -3 (x)

x x 0 u(s) ds → 0 when x → x along λ (x 0 , x). It is now an easy exercice by lemma 2.5 and (2.16) to see that min{|u|, |u |} → +∞ necessarily when x → x. (Just assume that u -1 (x) → 0 is false and see that there is a contradiction.)

End of the Proof of theorem 2.1. What remains to show is that x is a second order pole. The substitution u = 1/v 2 transforms (2.16) into

(v ) 2 = 1 + x 2 v 4 - v 6 2 x x 0 ds v 2 (s) + C 4 v 6 .
(2.30)

We know from lemma 2.6 that

v 6 2 x x 0 ds v 2 (s)
ds → 0 and v → 0 along a path λ (x 0 , x)

which avoids the poles of u in Chapter 3

D x 0 . Therefore (v ) 2 = 1 + o(1), then v 2 (x) = (x -x) 2 1 + o(1) . Plugging this last equality in (2.30) yields (v ) 2 (x) = 1 + x 2 (x -x) 4 + o((x -x) 4 ), thus v 2 (x) = (x -x) 2 + x 10 (x -x) 6 + o((x -x) 6

Tritruncated solutions for the first Painlevé equation

Abstract This chapter is devoted to the construction of the tritruncated solutions for the first Painlevé equation, the existence of which being announced in Sect. 2.6. This example will introduce the reader to common reasonings in resurgence theory. We construct a prepared form associated with the first Painlevé equation (Sec 3.1). This prepared ODE has a unique formal solution from which we deduce the existence of truncated solutions by application of the "'main asymptotic existence theorem".

We then study the Borel-Laplace summability property of the formal solution by various methods (Sect. 3.3). One deduces the existence of the tritruncated solutions for the first Painlevé equation, by Borel-Laplace summation (Sect. 3.4).

Normalization and formal series solution

Throughout this course, C[[z -1 ]] stands for the differential algebra of formal power series of the form g(z) = ∑ n≥0 a n z -n , while C((z -1 )) is the space of formal Laurent series. The space of formal Laurent series is a valuation field with the natural valuation val :

C((z -1 )) → Z ∪ ∞ ∑ n∈Z a n z -n → val w = min{n ∈ Z / a n = 0}. (3.1)

Normalization, prepared form

We saw in Sect. 2.6 that the first Painlevé equation is equivalent to the following differential equation, The variable z is most often called critical time [7].

v + v z = - 1 2 + 4 25 v z 2 + 1 2 v 2 , ( 3
It is worth mentioning that the symmetries detailed in Sect. 2.5 translate into the fact that any solution v of (3.2) is mapped into another solution v k through the transformation:

v k (z) = e iπk v e iπk/2 z , k = 0, • • • , 3. (3.3) 29
We look for a formal solution of (2.8) of the form v

(z) = ∞ ∑ l=0 b l z -l ∈ C[[z -1 ]].
When plugging this formal series in (3.2), one gets the necessary conditions: b 2 0 = 1, b 1 = 0 and b 2 = - 4 25 . Thanks to the symmetries (3.3), there is no restriction in assuming b 0 = 1. Also, it will be convenient in the sequel to make a new transformation,

v(z) = 1 - 4 25
1 z 2 + 1 z 2 w(z), (3.4)
which has the virtue of bringing (3.2) into the following differential equation :

w - 3 z w -w = 392 625 1 z 2 - 4 z 2 w + 1 2z 2 w 2 . ( 3
P(∂ )w + 1 z Q(∂ )w = F(z, w), with P(∂ ) = ∂ 2 -1, Q(∂ ) = -3∂ , ∂ = d dz (3.6)
and F(z, w) = 392 625

1 z 2 - 4 z 2 w + 1 2z 2 w 2 = f 0 (z) + f 1 (z)w + f 2 (z)w 2
, is called the prepared form equation associated with the first Painlevé equation.

Remark 3.1. For general comments on normalization procedures see, e.g. [7] and exercise 3.1. Notice that the prepared form is not uniquely defined.

Formal series solution

Substituting the formal series expansion ∞ ∑ l=0 a l z -l into equation (3.6) and identifying the powers, yields a quadratic recursion relation, namely:

           a 0 = a 1 = 0, a 2 = - 392 625
a l = l 2 a l-2 - 1 2 l-2 ∑ p=0 a (p) a (l-2-p) , l = 3, 4, • • • (3.7)
The following proposition is a simple exercise.

Proposition 3.1. There exists a unique formal series solution of (3.6) denoted by:

w(z) = ∞ ∑ l=0 a l z -l ∈ C[[z -1 ]]. (3.8)
Moreover the series w is even, val w = 2 and the coefficients a l are all real negative.

[[z -1 ]], N : C[[z -1 ]] → C[[z -1 ]]. When C[[z -1 ]
] is seen as a complete metric space (for the so-called Krull topology, see [19]), N appears as a contractive map and the formal solution w given by lemma 3.1 is the unique solution of the fixed point problem. This way of showing the existence of the formal solution w is also useful for numerical calculations,

w(z) = - 392 625 z -2 - 6272 625 z -4 - 141196832 390625 z -6 + • • •
In this course, all calculations have been produced that way under Maple 12.0 (released: 2008).

Towards truncated solutions

Notations

We fix notations (essentially common with [19,16]) which will be used in this chapter and throughout the course. and one denotes

• 0 (I ) • 0 (I) (resp. • ∞ (I ) • ∞ (I)) if the closure • ¯0(I ) (resp. • ¯∞(I )) is included in • 0 (I) (resp. • ∞ (I)).

Main asymptotic existence theorem

We have previously seen that the ODE (3.6) is formally solved by a unique formal series w(z

) ∈ C[[z -1 ]].
Question 3.1. Can we associate to w a holomorphic solution whose Poincaré asymptotics1 are governed by this formal series ? This question is the matter of the "main asymptotic existence theorem". This theorem is detailed in [16] for linear ODEs. It can be formulated to nonlinear equations, see [27], theorems 12.1 and 14.1, and [24] for extension to Gevrey asymptotics. Theorem 3.1 (Main asymptotic existence theorem M.A.E.T.). Let I ⊂ S 1 be an open arc of length |I| ≤ π/(q + 1) where q is a nonnegative integer. Let F(z, w) be a m-dimensional vector function subject to the following conditions:

1. F(z, w) is holomorphic in (z, w) on the domain of • ∞ (I) × B(0, r) with B(0, r) = {w ∈ C m , w ≤ r} for some r > 0;
2. F(z, w) admits an asymptotic expansion in powers of z -1 at infinity in • ∞ (I), uniformaly valid in w ∈ B(0, r); 3. the equation z -q w = F(z, w) is formally satisfied by a formal power series solution w(z) ∈ (C[[z -1 ]]) m ; 4. if F j (z, w) denotes the components of F(z, w), the Jacobian matrix

lim z→∞, z∈ • ∞ (I)    ∂ F 1 ∂ w 1 (z, 0) • • • ∂ F 1 ∂ w m (z, 0) • • • • • • • • • ∂ F m ∂ w 1 (z, 0) • • • ∂ F m ∂ w m (z, 0)    has non zero eigenvalues.
Then there exists a solution w of the equation z -q w = F(z, w), holomorphic in a domain of the form

• ∞ (I), whose asymptotics at infinity in every proper subsector of • ∞ (I) is given by the formal solution w.

Application

Let us transform (3.6) into a one order ODE of dimension 2. We introduce

w = w 1 w 2 = w w
and we obtain the companion system:

∂ w = 0 1 1 3 z w + 0 F(z, w 1 ) = F 1 (z, w) F 2 (z, w) = F(z, w) ∈ (C[z -1 , w]) 2 . (3.9)
We fix an open arc I ⊂ S 1 , arbitrary but of length |I| ≤ π. We also consider a domain of the form

• ∞ (I) and we make the following observations:

1. F(z, w) is polynomial with respect to w, with coefficients belonging to C[z -1 ].

Therefore F(z, w) is holomorphic in (z, w) on the domain 2 , F(z, w) admits an asymptotic expansion in powers of z -1 at infinity in • ∞ (I), uniformaly valid in w ∈ B(0, r); 3. the equation (3.9) is formally satisfied by a formal power series solution

• ∞ (I) × B(0, r) with B(0, r) = {w ∈ C 2 , w ≤ r} for some r > 0; 2. again because F(z, w) ∈ (C[z -1 , w])
w(z) = w w ∈ (C[[z -1 ]]) 2 ;
4. the Jacobian matrix 0 1

1 0 = ∂ F 1 ∂ w 1 (∞, 0) ∂ F 1 ∂ w 2 (∞, 0) ∂ F 2 ∂ w 1 (∞, 0) ∂ F 2 ∂ w 2
(∞, 0) has non zero eigenvalues

µ 1 = -1 and µ 2 = 1.
These properties allow to apply the (M.A.E.T.) and this shows the following proposition (see also [15]): As a matter of fact, from the above informations and the property for any solution of the first Painlevé equation to be a meromorphic function, one can even show the existence of tritruncated solutions [15]. However, to get more precise informations, we decide in what follows to turn to the question of the Borel-Laplace summability of w.

A reminder

We assume that the reader has a previous acquaintance with Borel-Laplace summation and a little background with resurgence theory, amply elaborated in [19,16] to which we refer. For the convenience of the reader, we offer a brief reminder of definitions and results used in this chapter.

Formal Borel transform and convolution product

Definition 3.4. The formal Borel transform B(z → ζ ) is the linear isomorphism B : C[[z -1 ]] → Cδ ⊕ C[[ζ ]] defined by g(z) = ∞ ∑ l=0 b l z -l → b 0 δ + g(ζ ), g(ζ ) = ∞ ∑ l=1 b l ζ l-1 Γ (l) .
The formal series g is the minor of g. The inverse map L = B -1 is the formal Laplace transform.

(b 0 δ + g) * (c 0 δ + h) = B( g h), where g = L (b 0 δ + g), h = L (c 0 δ + h). When g(ζ ) = ∑ n≥0 b n ζ n and h(ζ ) = ∑ n≥0 c n ζ n are two formal series, their convo- lution product g * h is given by the Hurwitz product, g * h(ζ ) = ∑ k≥1 d k ζ k with d k = ∑ n+m+1=k n!m! (n + m + 1)! b n c m . Proposition 3.3. The linear map ∂ : b 0 δ + g → -ζ g provides a derivation of Cδ ⊕ C[[ζ ]] and B : C[[z -1 ]], ∂ → Cδ ⊕ C[[ζ ]], ∂ is an isomorphism of dif- ferential algebras.
Gevrey series of order 1

Definition 3.6. A formal series g(z) = ∑ n≥0 a n z -n ∈ C[[z -1 ]] is 1-Gevrey when there exist constants C > 0, A > 0 so that |a n | ≤ C(n!)A n for all n. The space of 1-Gevrey series is denoted by C[[z -1 ]] 1 .
We recall from [16,19] that the space C[[z -1 ]] 1 of 1-Gevrey series is a differential algebra.

Notice that a formal series g is 1-Gevrey if and only if its minor g is a convergent power series, thus defines a germ of holomorphic functions (still denoted by g). More precisely:

Proposition 3.4. The restricted linear map B| : C[[z -1 ]] 1 → Cδ ⊕ O 0 is
an isomorphism of differential algebras. Also, for any two germs of holomorphic functions g, h ∈ O 0 , their convolution product g * h ∈ O 0 has the following integral represen-

tation: g * h(ζ ) = ζ 0 g(η) h(ζ -η)dη.
A flavor of resurgence Definition 3.7. Let Ω be a non-empty closed discrete subset of C and ϕ ∈ O 0 be a germ of holomorphic functions at 0. This germ is said to be Ω -continuable if there exists r > 0 such that D (0, r) ∩ Ω = / 0 and ϕ can be represented by a function holomorphic on D(0, r) which can be analytically continued along any path of C\Ω originating from any point of D (0, r). The space of all Ω -continuable germs is denoted by RΩ . The space Cδ ⊕ RΩ is called the space of Ω -resurgent functions. The space of Ω -resurgent formal series is denoted by RΩ and defined by RΩ = L Cδ ⊕ RΩ ).

Theorem 3.2. Let Ω 1 , Ω 2 be non-empty closed discrete subsets of C. Let Ω ⊂ C be the subset defined by Ω = Ω 1 ∪ Ω 2 ∪ (Ω 1 + Ω 2 )
where

Ω 1 + Ω 2 = {ω 1 + ω 2 | ω 1 ∈ Ω 1 , ω 2 ∈ Ω 1 }.
If Ω is closed and discrete, then

ϕ 1 ∈ RΩ 1 and ϕ 2 ∈ RΩ 2 imply ϕ 1 * ϕ 2 ∈ RΩ .
In particular, the space Cδ ⊕ RZ of Z-resurgent functions is stable under convolution product, thus is an algebra with unit δ .

Borel-Laplace summability

Definition 3.8. A formal series g(z) = ∑ n≥0 b n z n ∈ C[[z -1 ]
] is said to be Borel-Laplace summable in direction θ ∈ S 1 if the following conditions are satisfied:

• the series g is 1-Gevrey or, equivalently, its minor g is a convergent series whose sum defines a holomorphic function (still denoted by g) near the origin ;

• g can be analytically continued to an open sector of the form

• ∞ 0 (I) where I ⊂ S 1 is an open neighbourdhood of θ , with exponential growth of order 1 at infinity. Under the above conditions, the Borel-Laplace sum of g in direction θ is denoted by S θ g and defined by S θ g(z) = L θ • B g(z) where L θ stands for the Laplace

transform in direction θ , L θ (b 0 δ + g)(z) = b 0 + ∞e iθ 0 e -zζ g(ζ )dζ .
In addition to this definition, we recall that the Borel-Laplace sum S θ w is holomorphic on a half-plane where its asymptotic behavior is governed by the formal series g. This will be made more precise in a moment.

Formal series solution and Borel-Laplace summability

We go back to the formal series w given by proposition 3.1. Since val w > 0, the formal Borel transform of w just reduces to its minor w. Also, w(z) is the unique solution in C[[z -1 ]] of the differential equation (3.6). One easily infers the following result from the general properties of the formal Borel transform.

P( ∂ ) w + 1 * Q( ∂ ) w = f 0 + f 1 * w + f 2 * w * w, P( ∂ ) = ∂ 2 -1, Q( ∂ ) = -3 ∂ , f 0 (ζ ) = 392 625 ζ , f 1 (ζ ) = -4ζ , f 2 (ζ ) = 1 2 ζ .
(3.10)

We will see in a moment that w is 1-Gevrey and even Borel-Laplace summable. In the rest of this chapter, we analyse this Borel-Laplace summability and we offer various approaches.

Formal series solution and Borel-Laplace summability: a perturbative approach

We start with a perturbative approach which has the advantage of giving a first insight into the resurgent structure. In practice, we consider (3.10) as a perturbation of the equation P( ∂ ) w = f 0 which is quite easy to solve:

• either formally since the map P( ∂ ) :

g ∈ C[[ζ ]] → (ζ 2 -1) g ∈ C[[ζ ]] is invertible;
• or analytically, in a space of analyic functions, say O 0 , because

P( ∂ ) : g ∈ O 0 → (ζ 2 -1) g ∈ O 0 is once again invertible.
To keep one, it is convenient to transform equation (3.10) into the following one parameter family of convolution equations,

P( ∂ ) h = f 0 + ε -1 * Q( ∂ ) h + f 1 * h + f 2 * h * h , (3.11)
and to look for a solution under the form

h(ζ , ε) = ∑ l≥0 h l (ζ )ε l . (3.12)
When plugging (3.12) into (3.11) and identifying the same powers in ε, one obtains a recursive system of convolution equations, namely:

               P( ∂ ) h 0 = f 0 , P( ∂ ) h 1 = -1 * Q( ∂ ) h 0 + f 1 * h 0 + f 2 * h 0 * h 0 , P( ∂ ) h n = -1 * Q( ∂ ) h n-1 + f 1 * h n-1 + ∑ n 1 +n 2 =n-1 f 2 * h n 1 * h n 2 , n ≥ 1.
(3.13) Proof. Use the fact that the map P( ∂

) : C[[ζ ]] → C[[ζ ]
] is invertible and the general properties of the convolution product.

The above lemma has the following consequence: We mention that proposition 3.6 has a counterpart by formal Laplace transform L (ζ → z). Introducing h l = L h l , one gets from lemma 3.1 that the sequence ( h l ) l≥0 solves in C[[z -1 ]] the following recursive system of linear nonhomogeneous ODEs:

                   P(∂ ) h 0 = f 0 (z) P(∂ ) h 1 = - 1 z Q(∂ ) h 0 + f 1 (z) h 0 + f 2 (z) h 2 0 P(∂ ) h n = - 1 z Q(∂ ) h n-1 + f 1 (z) h n-1 + f 2 (z) ∑ n 1 +n 2 =n-1 h n 1 h n 2 , n ≥ 1. (3.14)
From lemma 3.1 again, one deduces that h l ∈ z -2l-2 C[[z -1 ]] for every l ≥ 0, thus:

Proposition 3.7. The series ∑ l≥0 h l (z) is well defined in C[[z -1 ]
] and is formally convergent to the unique formal solution w(z) ∈ C[[z -1 ]] of the differential equation (3.6).

Analytic properties and a flavor of resurgence

Instead of working in the space of formal series, one can rather work in a space of analytic functions. The next proposition uses definition 3.7.

Proposition 3.8. For every l ∈ N, the formal series h l given by (3.13) defines a germ (still denoted by h l ) of holomorphic functions at 0, which can be represented by a function holomorphic on the open disc D(0, 1). Moreover, h l belongs to the space RΩ l of Ω l -resurgent functions, where Ω l = {0, ±1, • • • , ±l, ±(l + 1)}. As a consequence, the germ h l is a Z-resurgent function.

Proof. The proposition is easily shown by induction from (3.13), theorem 3.2 and the following remark : for every l ∈ N, RΩ l ⊂ RΩ l+1 and the linear map

P( ∂ ) : g ∈ RΩ l → (ζ 2 -1) g ∈ RΩ l is invertible.

Further preparations

We have previously seen (proposition 3.6) that the minor w of the formal series w solution of the prepared form equation (3.6), can be written as

w(ζ ) = ∑ l≥0 h l (ζ ) in the space C[[ζ ]],
where the sequence ( h l ) l≥0 solves the recursive system of equations (3.13). To show the Borel-Laplace summability of w, it is thus enough to check the following properties:

• the series of functions ∑ l≥0 h l (ζ ) converges to a holomorphic function near the origin and can be analytically continued in a convenient sector; • this function has at most exponential growth of order 1 at infinity in this sector.

We also know by proposition 3.8 that each h l (ζ ) is a Z-resurgent function. This motivates the following definition, with the notations : D(a, r) is the open disc centred in a with radius r and D(a, r) is its closure. Definition 3.9. One sets Notice that | f | is also analytic on D(0, r).

D (0) ρ = λ =±1 D(λ , ρ) for any ρ ∈]0, 1[. We denote by • R (0) ρ the star-shaped domain defined by: • R (0) ρ = C \ tζ | t ∈ [1, +∞[, ζ ∈ D(±1, ρ) ⊂ C \ D (0) ρ , and • R (0) = 0<ρ<1 • R (0)
Lemma 3.2. Let be ρ ∈]0, 1[. There exists a constant M ρ,(0) > 0 such that for every

polynomial q ∈ C[ζ ] of degree ≤ 1 and every ζ ∈ C \ D (0) ρ , one has q(ζ ) P(-ζ ) ≤ M ρ,(0) |q|(1). Moreover, on can choose M ρ,(0) = 1 ρ . Proof. By definition of D (0) ρ , 1 |ζ ± 1| ≤ 1 ρ and ζ ζ ± 1 ≤ 1+ 1 ρ for every ζ ∈ C \ D (0) ρ .
Therefore,

ζ p P(-ζ ) ≤ 1 ρ 2-p 1 + 1 ρ p ≤ 1 ρ 2 (ρ + 1) p ≤ 2 p
ρ 2 for p = 0, 1, 2. This means that one can choose M ρ,(0) = 2 ρ 2 in the lemma. It is possible to be more As a rule, we will combined lemma 3.2 with the following lemma whose proof is left as an exercise (see [19,16]):

Majorant functions

We have in mind to show that the series of functions ∑ l≥0 h l (ζ ), discussed in propositions 3.6 and 3.8, is uniformaly convergent on any compact subset of • R (0) . We will use majorant functions which we now define.

Definition of the majorant functions We consider, for any ρ ∈]0, 1[, the sequence of functions ( H l ) l≥0 recursively defined by:

                       1 M ρ,(0) H 0 = | f 0 |(ξ ), 1 M ρ,(0) H 1 = 3 + | f 1 | * H 0 + | f 2 | * H 0 * H 0 , 1 M ρ,(0) H n = 3 + | f 1 | * H n-1 + ∑ n 1 +n 2 =n-1 | f 2 | * H n 1 * H n 2 , n ≥ 1. (3.15)
where M ρ,(0) is given by lemma 3.2 and (3.15) with (3.13).) We claim that for every l ∈ N, H l is a majorant function for h l . Precisely: Lemma 3.4. For every ρ ∈]0, 1[ and every l ∈ N, the following properties are satisfied: H l (ξ ) is a polynomial which belongs to ξ l+1 C[ξ ]; furthermore,

| f 0 |(ξ ) = 392 625 ξ , | f 1 |(ξ ) = 4ξ , | f 2 |(ξ ) = 1 2 ξ . (Compare
for every ζ ∈ • R (0) ρ , h l (ζ ) ≤ H l (ξ ) with ξ = |ζ |, (3.16)
where ( h l ) l≥0 is defined by (3.13).

Proof. The fact that H l (ξ ) ∈ ξ l+1 C[ξ ] is proved by induction from (3.15) and the properties of the convolution product. By (3.13) and lemma 3.2, for every ζ ∈

• R (0) ρ , h 0 (ζ ) ≤ 1 P(-ζ ) f 0 (ζ ) ≤ M ρ,(0) | f 0 |(ξ ) with ξ = |ζ |, so that (3.16
) is true for l = 0. We now assume that (3.16) is true for l = 0, • • • , (n -1), for some n ∈ N . By lemma 3.3 and the induction hypothesis, for every ζ ∈

• R (0) ρ , 1 P(-ζ ) . 1 * Q( ∂ ) h n-1 (ζ ) ≤ 1 P(-ζ ) |Q|(|ζ |) 1 * H n-1 (|ζ |) ,
where |Q|(ξ ) = 3ξ . Therefore, by lemma 3.2,

1 P(-ζ ) . 1 * Q( ∂ ) h n-1 (ζ ) ≤ M ρ,(0) |Q|(1) 1 * H n-1 (ξ )
with ξ = |ζ |. More generally, for similar reasons,

1 M ρ,(0) h n (ζ ) ≤ 3 * H n-1 (ξ ) + | f 1 | * H n-1 (ξ ) + ∑ n 1 +n 2 =n-1 | f 2 | * H n 1 * H n 2 (ξ ).
Thus, for every ζ

∈ • R (0) ρ , h n (ζ ) ≤ H n (ξ )
. This ends the proof. Upper bounds for the majorant functions Before keeping on studying the above majorant functions, we state a property which will be useful in the sequel. We first recall two notations.

For R 0 > 0, we set D(∞, R 0 ) = z ∈ C, |z| > 1 R 0 . Lemma 3.5. Let be R 0 > 0. We suppose f ∈ O D(∞, R 0 ) with f (z) = O(z -m
) at infinity for a certain m ∈ N, and let be M = sup

z∈D(∞,R 0 ) | f (z)|.
Then the formal Borel transform B f = f 0 δ + f of f satisfies the following properties:

1. f ∈ O(C) and | f 0 | ≤ M R 0 . 2. for every ζ ∈ C, | f (ζ )| ≤ f (ξ ) ≤ M R 0 e ξ R 0 with ξ = |ζ | and, when m ≥ 2, | f (ζ )| ≤ M R m 0 ξ m-2 (m -2)! * e ξ R 0 , ξ = |ζ |.
Proof. The Taylor series expansion of f , ∑

k≥m f k z -k = z -(m-1) ∑ l≥1 f m+l-1 z -l , con- verges to f in D(∞, R 0 ). By the Cauchy inequalities, | f k | ≤ M R k 0 for any k ∈ N. The formal Borel transform of f reads B f = f 0 δ + f with : 1. f (ζ ) = ∑ l≥1 f l ζ l-1 (l -1)! as rule, 2. f (ζ ) = ζ m-2 (m -2)! * ∑ l≥1 f m+l-1 ζ l-1 (l -1)! when m ≥ 2. Also, for every ζ ∈ C, ∑ l≥1 | f m+l-1 | |ζ | l-1 (l -1)! ≤ ∑ l≥1 M R m+l-1 0 ξ l-1 (l -1)! ≤ M R m 0 e ξ R 0 with ξ = |ζ |.
This ensures the uniform convergence on any compact set of C, thus f ∈ O(C), and provides the upper bounds.

We return to the majorant functions defined by (3.15).

Lemma 3.6. For every l ∈ N, the majorant function H l (ξ ) is the formal Borel transform of H l (z) which has the following properties: H l (z) belongs to C[z -1 ] and, for every ρ ∈]0, 1[, H l (z) is bounded on the domain |z| > 8 ρ , precisely sup

|z|> 8 ρ | H l (z)| ≤ 1 2 l .
Proof. We introduce the generating function:

H(ξ ) = ∞ ∑ l=0 H l (ξ ) ε l ∈ C[ξ ][[ε]]. From (3.
H = | f 0 | + ε 3 + | f 1 | * H + | f 2 | * H * H . (3.17)
Therefore, H can be seen as the formal Borel transform of the solution

H(z, ε) = ∞ ∑ l=0 H l (z) ε l ∈ C[z -1 ][[ε]
] of the following second order algebraic equation:

1 M ρ,(0) H = | f 0 |(z) + ε 3 z + | f 1 | H + | f 2 | H 2 with | f 0 |(z) = 392 625 1 z 2 , | f 1 |(z) = 4 z 2 , | f 2 |(z) = 1 2z 2 . (3.18)
This equation has two branch solutions and one of them is asymptotic to the equation 1 M ρ,(0) H = | f 0 | when ε goes to zero. We are interested in that solution. Instead of using an explicit calculation, we rather use another method which can be generalized. In (3.18) we make the change of variable t = 1 z and set H(z, ε) = H(t, ε).

The equation (3.18) becomes:

F (t, ε, H) = 0, with (3.19)
F (t, ε, H) = 1 M ρ,(0) H -| f 0 |(t -1 ) -ε 3t + | f 1 |(t -1 ) H + | f 2 |(t -1 )H 2 .
Since F (0, 0, 0) = 0 and

∂ F ∂ H (0, 0, 0) = 1 M ρ,(0)
= 0, the implicit function theorem provides a unique holomorphic solution H(t, ε) to (3.19), for |t| and |ε| small enough : there exist r 1 > 0, r 2 > 0, r 3 > 0 and a holomorphic function

H : (t, ε) ∈ D(0, r 1 ) × D(0, r 2 ) → H(t, ε) ∈ D(0, r 3 ) such that for every (t, ε, H) ∈ D(0, r 1 ) × D(0, r 2 ) × D(0, r 3 ), F (t, ε, H) = 0 ⇔ H = H(t, ε) .
To get more precise informations, we view the implicit problem (3.19) as a fixedpoint problem:

H = N(H), (3.20)
N(H) = M ρ,(0) | f 0 |(t -1 ) + ε 3t + | f 1 |(t -1 ) H + | f 2 |(t -1 )H 2 = M ρ,(0) 392 625 t 2 + ε 3t + 4t 2 H + 1 2 t 2 H 2 .
We choose M ρ,(0) = 1 ρ (see lemma 3.2) and we introduce the space O(U) of functions in (t, ε) which are holomorphic on the polydisc U = D(0, ρ 8 ) × D(0, 2) and continuous on the closure U. The space O(U), is a Banach algebra where stands for the maximum norm.

We recall the following theorem [26]: let U be a bounded open subset of C n , n ≥ 1, E be a Banach space and O(U) be the space of functions f : x → f (x) ∈ E which are continuous on U and holomorphic on U. With the the maximum norm f = sup

z∈U | f (z)|, (O(U), . ) is a Banach algebra.
For a reason of homogeneity, we introduce the ball

B ρ = {H ∈ O(U), H ≤ ρ}. For any H, H 1 , H 2 ∈ B ρ , N(H) ≤ 1 ρ 392 625 ρ 2 64 + 2 7ρ
16 H + ρ 2 128 H 2 ≤ ρ (remember that ρ < 1), while

N(H 1 ) -N(H 2 ) ≤ 2 ρ 7ρ 16 H 1 -H 2 + ρ 2 128 H 1 -H 2 H 1 + H 2 ≤ 29 32 H 1 -H 2 .
The mapping

N |B ρ : H ∈ B ρ → N(H) ∈ B ρ is thus contractive. Since B ρ is a closed subset of a complete space, (B ρ , .
) is complete and the contraction mapping theorem can be applied. We deduce the existence of a unique solution H in B ρ of the fixed-point problem (3.20). This solution H(t, ε), thus holomorphic in U = D(0, ρ 8 ) × D(0, 2), has a Taylor expansion with respect to ε at 0 of the form

H(t, ε) = ∞ ∑ l=0 H l (t) ε l
, where (H l ) l≥0 is a sequence of holomorphic functions on the disc D(0, ρ 8 ). Moreover, by the Cauchy inequalities and using the fact that sup

(t,ε)∈U |H(t, ε)| ≤ ρ, one gets: for every l ∈ N, sup t∈D(0, ρ 8 ) |H l (t)| ≤ ρ 2 l
. This ends the proof of lemma 3.6. Proof. This is due to lemmas 3.5 and 3.6.

Formal series solution and Borel-Laplace summability

We are ready to show the following theorem. Proof. Combining lemmas 3.4 and 3.7, we know that, for every ρ ∈]0, 1[ and l ≥ 0,

the function h l (ζ ), is holomorphic on • R (0) ρ . Moreover, for every R > 0, setting U R = D(0, R) ∩ • R (0) ρ , ∑ l≥0 sup U R | h l (ζ )| ≤ ∑ l≥0 H l (R) ≤ ∑

Formal series solution and Borel-Laplace summability: second approach

In this second approach, however related to the first one, we introduce a Banach space (following [6,7]), convenient to analyse the analyticity of the formal Borel transform of the formal series w solution of the ODE (3.6). We then introduce the reader to a "Grönwall-like lemma" which will give the upper bounds we are looking for.

∂ : f ∈ O(U) → ∂ f (ζ ) = -ζ f (ζ ) ∈ M O(U).
Let be ν ≥ 0. The norm . ν is defined as follows: for every f ∈ O(U),

f ν = R sup ζ ∈U e -ν|ζ | f (ζ ) .
This norm is extended to Cδ ⊕ O(U) by setting: cδ

+ f ν = |c| + f ν , while ∂ δ = 0.
Proposition 3.9. The space Cδ ⊕ O(U), . ν is a Banach algebra. In particular, for every f , g

∈ Cδ ⊕ O(U), f * g ν ≤ f ν g ν . The space M O(U) is closed in the normed space O(U), . ν . Moreover, for ν > 0:
1. for every n ∈ N, for every g ∈ O(U),

ζ n * g ν ≤ n! ν n+1 g ν , (ζ → ζ n+1 ) ν ≤ n! ν n+1 R and (ζ → 1) ν = R. 2. for every f , g ∈ O(U), f g ν ≤ 1 R f ν g 0 . 3. for every f ∈ O(U), ν ≥ ν 0 ≥ 0 ⇒ f ν ≤ f ν 0 . 4. for every f ∈ M O(U), lim ν→∞ f ν = 0. 5. the derivation ∂ | O(U) : f ∈ O(U) → ∂ f ∈ M O(U) is invertible. Its inverse map ∂ -1 satisfies: for every f ∈ O(U), for every g ∈ M O(U), ∂ -1 ( f * g) ∈ M O(U) and ∂ -1 ( f * g) ν ≤ 1 νR f ν ∂ -1 g 0 . Also, for every f ∈ Cδ ⊕O(U), for every g ∈ M O(U), ∂ -1 ( f * g) ∈ O(U) and ∂ -1 ( f * g) ν ≤ f ν ∂ -1 g ν . Proof. Since Re -νR sup ζ ∈U f (ζ ) ≤ R sup ζ ∈U e -ν|ζ | f (ζ ) ≤ R sup ζ ∈U f (ζ ) , we see that
. ν is equivalent to the usual maximum norm on the vector space O(U) and this normed vector space is complete. This shows the completeness of O(U), +, . , . Therefore

R|e -ν|ζ | f * g(ζ )| ≤ f ν g ν |ζ | 0 1 R ds ≤ f ν g ν . We conclude that for every f , g ∈ O(U), f * g ν ≤ f ν g ν , hence O(U), .
ν is a Banach algebra and Cδ ⊕ O(U), . ν as well.

We now suppose ν > 0.

1. For the particular case f : ζ → ζ n and g ∈ O(U): This shows that

Re -ν|ζ | (ζ n * g)(ζ ) ≤ R
ζ n * g ν ≤ n! ν n+1 g ν . The other properties follow. 2. Obviously, f g ν ≤ f ν sup U |g| ≤ 1 R f ν g 0 , for every f , g ∈ O(U). 3. It is straightforward to see that ν ≥ ν 0 ≥ 0 implies f ν ≤ f ν 0 when f ∈ O(U). 4. If f ∈ M O(U), then f = ζ g with g ∈ O(U).
From the previous property,

f ν ≤ 1 R ζ ν g 0 ≤ 1 ν g 0 . Thus lim ν→∞ f ν = 0. 5. If f ∈ Cδ ⊕ O(U) and g ∈ M O(U) then f * g ∈ M O(U). Assume now that f ∈ O(U) and g ∈ M O(U). Then ∂ -1 ( f * g)(0) = 0 and writing ζ = |ζ |e iθ ∈ U, Re -ν|ζ | f * g(ζ ) = Re -ν|ζ | |ζ | 0 g(se iθ ) f (|ζ | -s) e iθ ds (3.21) = R |ζ | 0 se iθ ( ∂ -1 g)(se iθ )e -νs f (|ζ | -s)e iθ e -ν(|ζ |-s) e iθ ds.
On the one hand, from (3.21),

R|e -ν|ζ | f * g(ζ )| ≤ 1 R f ν ∂ -1 g ν |ζ | 0 s ds ≤ |ζ | 2 2R f ν ∂ -1 g ν , so that R|e -ν|ζ | ∂ -1 ( f * g)(ζ )| ≤ |ζ | 2R f ν ∂ -1 g ν ≤ f ν ∂ -1 g ν . Thus ∂ -1 ( f * g) ν ≤ f ν ∂ -1 g ν .
One easily extends this formula to the case f ∈ Cδ ⊕ O(U). On the other hand, from (3.21),

R|e -ν|ζ | f * g(ζ )| ≤ f ν sup U | ∂ -1 g| |ζ | 0 se -νs ds ≤ |ζ | νR f ν ∂ -1 g 0 , hence R|e -ν|ζ | ∂ -1 ( f * g)(ζ )| ≤ 1 νR f ν ∂ -1 g 0 , . Therefore: ∂ -1 ( f * g) ν ≤ 1 νR f ν ∂ -1 g 0 .
This ends the proof.

A Grönwall-like lemma

We start with the following observation.

w(z) = d z + a z + b z 2 w(z)+c N ∑ n=0 F n (z) w n (z), (3.23)
where the (F n ) 0≤n≤N is a (N + 1)-tuple of holomorphic functions on a neighbourhood of infinity with F n (z) = O(z -1 ). This shows (by a reasoning already done) that

w d = O(z -1
) is a holomorphic function in (z, d) for d ∈ C and z on a neighbourhood of infinity (independent on d). Therefore, w d determines a function holomorphic in

(ξ , d) ∈ C 2 ,
ζ ∈U |q|(|ζ |) |p(-ζ )| < ∞, b = sup ζ ∈U |r|(|ζ |) |p(-ζ )| < ∞, c = sup ζ ∈U 1 |p(-ζ )| < ∞.
We finally assume that w ∈ O(U) solves the following convolution equation: Proof. (Adapted from [17]). We assume that w ∈ O(U) is a solution of the convolution equation (3.22). We thus have, for every ζ ∈ U, Assume on the contrary that there exists

p( ∂ ) w + 1 * [q( ∂ ) w] = ζ * [r( ∂ ) w] + f 0 + N ∑ n=1 f n * w * n . ( 3
p( ∂ ) w(ζ ) = f 0 (ζ ) - ζ 0 [q( ∂ ) w](η) dη + ζ 0 (ζ -η)[r( ∂ ) w](η) dη + N ∑ n=1 ζ 0 f n (ζ -η) w * n (η) dη Thus, writing ξ = |ζ | and ζ = ξ e iθ , | w(ζ )| ≤ 1 |p(-ζ )| F 0 (ξ ) + ξ 0 |q|(ξ ) |p(-ζ )| + |r|(ξ ) |p(-ζ )| (ξ -r) | w(re iθ )| dr + N ∑ n=1 ξ 0 1 |p(-ζ )| F n (ξ -r)| w * n (re iθ )| dr.
ζ 1 = ξ 1 e iθ ∈ U such that | w(ζ 1 )| ≥ w d (ξ 1 ). Define χ = {ζ ∈ [0, ζ 1 ] | | w(ζ )| ≥ w d (|ζ |)}
. This is a non-empty closed set, bounded from below, and we note ζ 2 its infimum. Putting things together, one gets with

• If | w(ζ )| ≥ w d (|ζ |) for some ζ ∈]0, ζ 2 [, then ζ ∈ χ and this contradicts the definition of ζ 2 . Thus, for every ζ ∈ [0, ζ 2 [, | w(ζ )| < w d (|ζ |). • If | w(ζ 2 )| > w d (|ζ 2 |) then,
ξ 2 = |ζ 2 |: | w(ζ 2 )| ≤ c F 0 (ξ 2 ) + ξ 2 0 [a + b(ξ 2 -r)] | w(re iθ )| dr + c N ∑ n=1 ξ 2 0 F n (ξ 2 -r)| w * n (re iθ )| dr ≤ c F 0 (ξ 2 ) + ξ 2 0 [a + b(ξ 2 -r)] w d (r) dr + c N ∑ n=1 ξ 2 0 F n (ξ 2 -r) w * n d (r) dr.

Applications

We prove theorem 3.3 with the tools introduced in this section. For R > 0 and ρ > 0, we introduce the star-shaped domain

U R = D(0, R) ∩ • R (0) ρ . We set B r = { v ∈ O(U R ), v ν ≤ r}, r > 0 and ν > 0.
We consider the convolution equation (3.10), viewed as a fixed-point problem. Precisely, we consider the mapping

N : v ∈ B r → P( ∂ ) -1 -1 * Q( ∂ ) v + f 0 + f 1 * v + f 2 * v * v .
By lemmas 3.2 and proposition 3.9, one first gets:

N( v) ν ≤ M ρ,(0) -1 * Q( ∂ ) v + f 0 + f 1 * v + f 2 * v * v ν .
By proposition 3.9 again, since Q( ∂ ) = -3 ∂ , one easily obtains:

1 * Q( ∂ ) v ν ≤ 1 ν Q( ∂ ) v ν ≤ 1 Rν Q(-ζ ) 0 v ν ≤ 3 ν v ν .
The functions f 0 , f 1 , f 2 belong to M O(U R ). By proposition 3.9, this implies lim ν→∞ f i ν = 0, i = 0, 1, 2. We then deduce N( v) ν ≤ r by choosing ν > 0 large enough. By the same arguments, one easily sees that

N( v 1 ) -N( v 2 ) ν ≤ k v 1 -v 2 ν with k < 1, for v 1 , v 2 ∈ B r
and for ν > 0 large enough. This means that N is contractive in the closed set B r of the Banach space O(U R ), . ν , for ν > 0 large enough. The contraction mapping theorem provides a unique solution w ∈ B r for the fixed-point problem v = N( v). Since R and ρ can be arbitrarily chosen, we deduce (by uniqueness) that the formal Borel transform w of the unique formal series w solution of (3.6), defines a holomorphic in

w = | f 0 | + 3 + | f 1 | * w+| f 2 | * w * w.
This is nothing but (3.17) with ε = 1. We adopt the notations and reasoning made for the proof of lemma 3.6. Let w(z) be the inverse Borel transform of w and w(z) = H(t), t = z -1

First Painlevé equation and tritruncated solutions

Theorem 3.3 shows that one can apply the Borel-Laplace summation scheme to the unique formal series expansion w ∈ C[[z -1 ]] solving equation (3.6). This is what we do in this section which starts with a brief reminder.

Reminder

We complete definitions 3.3 and definition 3.8 with notations essentially common with [19,16]. For the convenience of the reader we also recall some results about Borel-Laplace summability and we refer to [19,16] for more details. Let

g = ∑ n≥0 b n z n ∈ C[[z -1 ]
] 1 be a 1-Gevrey series: the minor g thus determines a holomorphic function near the origin (still denoted by g). We add the following conditions:

• one can find an open arc I ⊂ S 1 such that g can be analytically continued to an open sector of the form

• ∞ 0 (I);
• this function (still denoted by) g is of exponential growth of order 1 at infinity: for every proper-subsector

• ∞ (I ) • ∞ 0 (I), there exist A > 0 and τ > 0 such that for every ζ ∈ • ∞ (I ), | g| ≤ Ae τ|ζ | .
Under these conditions, for every direction θ ∈ I , the Borel-Laplace sum S θ g is well-defined and holomorphic on the half-plane • Π θ τ . Moreover, for two close directions θ 1 , θ 2 ∈ I , the Borel-Laplace sums S θ 1 g and S θ 2 g coincide on their common domain

• Π θ 1 τ ∩ • Π θ 2
τ , thus can be glued together to give a holomorphic function on Fine Borel-Laplace summations It is possible to get more precise estimates than those given by (3.26), by appealing to fine Borel-Laplace summations, discussed with much attention in [16,19] to whom we refer. The following proposition is the easy part of a theorem due to Nevanlinna [19,16,18,12,25].

• Π θ 1 τ ∪ • Π θ 2 τ . More generally: Proposition 3.10. Let g(z) = ∑ n≥0 b n z n ∈ C[[z -1 ]] 1 be
Proposition 3.13. Let ϕ(z) = +∞ ∑ n=0 a n z n ∈ C[[z -1 ]
] 1 be a 1-Gevrey series, r > 0, A > 0, τ > 0 and θ a direction. Then property (1) implies property ( 2) in what follows.

1. The minor ϕ is analytically continuable on S r (θ ) and for every

ζ ∈ S r (θ ), | ϕ(ζ )| ≤ Ae τ|ζ | . 2. The Borel-Laplace sum S θ ϕ(z) is holomorphic in • Π θ τ and for every p ≥ 0, N ≥ 0 and z ∈ • Π θ τ : d p S θ ϕ dz p (z) - N ∑ k=p (-1) p a k-p (k -p) • • • (k -1) z k ≤ R as (r, A, τ, N, ze iθ ; p) (3.28) where R as (r, A, τ, N, z; p) = A N!e τr r N |z| N p! (ℜ(z) -τ) p+1 p ∑ l=0 r(ℜ(z) -τ) l l! (3.29)
Applications We return to theorem 3.3 and corollary 3.1. We consider a direction θ ∈]0, π[ and we choose r > 0 and 0 < ρ < 1 such that sin(θ ) = r + ρ. This ensures that the half-strip S r (θ ) is a subset of the domain

=]0, π[, resp. I 1 =]π, 2π[. The Borel-Laplace sum w tri,0 = S ]0,π[ w, resp w tri,1 = S ]π,2π[ w.
is a holomorphic solution of the differential equation (3.6) and w tri,0 , w tri,1 satisfy the following properties. For every θ ∈ I 0 , resp. θ ∈ I 1 , for every r > 0 and ρ > 0 so that | sin(θ )| = r + ρ, there exist τ > 0 and A > 0 such that :

• w tri, j ∈ O( • Π θ τ ), j = 0 resp. j = 1; • for every z ∈ • Π θ τ , for every N ∈ N, for j = 0 resp. j = 1, w tri, j (z) - N ∑ k=0 a k z k ≤ A N!e τr r N |z| N 1 ℜ(ze iθ ) -τ ; (3.30) dw tri, j dz (z) + N ∑ k=1 (k -1)a (k-1) z k ≤ A N!e τr r N |z| N 1 + r(ℜ(ze iθ ) -τ) ℜ(ze iθ ) -τ 2 (3.31)
where the coefficients a k are given by (3.7);

• morover one can choose A = 4.22, τ = 4.22 ρ . In particular w tri,0 , resp. w tri,1 , is holomorphic in

• D(I0, τ), resp. in • D(I1, τ), with τ(θ ) = 4.22 | sin(θ )| .

A link with 1-summability theory

We assume that the reader has a previous acquaintance with 1-summability theory, introduced and much discussed in [16], to which we refer. We only fix some notations, these are classical [16,18] but for the fact that we consider asymptotics at infinity, and we recall some properties.

Definition 3.17. Let I ⊂ S 1 be an open arc and

• ∞ = • ∞ (I) a sector. 1. A ( • ∞ ), resp. A (I),
is the differential algebra of holomorphic functions on the sector • ∞ admitting Poincaré asymptotics at infinity in this sector, resp. asymptotics germs at infinity over I.

The linear map

T : A ( • ∞ ) → C[[z -1 ]], resp. T : A (I) → C[[z -1 ]], which assigns to each f ∈ A ( • ∞ ), resp. f ∈ A (I)
, its asymptotic expansion at infinity, is called the Taylor map.

The Taylor map T is a morphism of differential algebras and this map is onto (Borel-Ritt theorem).

A 1 (

• ∞ ), resp. A 1 (I), is the differential algebra of holomorphic functions on the sector • ∞ with 1-Gevrey asymptotics at infinity in this sector, resp. 1-Gevrey asymptotics germs at infinity over I.

On denotes by T

1 : A 1 ( • ∞ ) → C[[z -1 ]] 1 , resp. T 1 : A 1 (I) → C[[z -1 ]] 1 , the Taylor map restricted A 1 ( • ∞ ), resp. A 1 (I), called the 1-Gevrey Taylor map.
The 1-Gevrey Taylor map T 1 is morphism of differential algebras. This map is onto when |I| ≤ π (Borel-Ritt theorem). This map is injective when |I| > π (Watson lemma).

A

<0 ( • ∞ ), resp. A <0 (I), is the space of flat functions on • ∞ , resp. flat germs at infinity over I. A <0 ( • ∞ ) is thus the kernel of the Taylor map T : A ( • ∞ ) → C[[z -1 ]] 4. A ≤-1 ( • ∞ ), resp. A ≤-1 (I)
, is the space of 1-exponentially flat functions on • ∞ , resp. 1-exponentially flat germs at infinity over I.

A ≤-1 ( • ∞ ) is the kernel of the 1-Gevrey Taylor map T 1 : A 1 ( • ∞ ) → C[[z -1 ]] 1 .

5.

A is the sheaf over S 1 of asymptotic functions at infinity associated with the presheaf A . We denote by A 1 the sheaf over S 1 of 1-Gevrey asymptotic functions at infinity associated with the presheaf A 1 . We denote by A <0 the sheaf over S 1 of flat germs at infinity associated with the presheaf A <0 . Finally A ≤-1 stands for the sheaf over S 1 of 1-Gevrey flat germs at infinity associated with the presheaf A ≤-1 .

Theorem 3.4 (Borel-Ritt). The quotient sheaf A /A <0 , resp. A 1 /A ≤-1 , is isomorphic via the Taylor map T , resp. the 1-Gevrey Taylor map T

1 , to the constant sheaf, C[[z -1 ]] resp. C[[z -1 ]] 1
We now go back to proposition 3.14. On the one hand, The domain

• D(I0, τ)
is a sectorial neighbourhood of ∞ with aperture

( I 0 =] - 3 2 π, + 1 2 π[. On the other hand, while • D(I1, τ) = e -iπ • D(I0, τ) is a sectorial neighbourhood of ∞ with aper- ture ( I 1 =] - 5 2 π, - 1 2 π[. These two open arcs provide a good covering { ( I 0 , ( I 1 } of the circle of directions S 1 . Let J 0 =] - 1 2 π, 1 2 π[ and J 1 =] - 3 2 π, - 1 2
π[ be the two intersection arcs. Both w tri,0 and w tri,1 can be considered as defining sections of A 1 , namely w tri,0 ∈ Γ ( ( I 0 , A 1 ) and w tri,1 ∈ Γ ( ( I 1 , A 1 ), and are asymptotic to the same 1-Gevrey formal series w. The pair (w tri,0 , w tri,1 ) defines a 0-cochain in the sense of Čech cohomology, and the 1-coboundary (w tri,0w tri,1 , w tri,1w tri,0 ) belongs to Γ (J 0 , A ≤-1 ) × Γ (J 1 , A ≤-1 ).

Miscellaneous properties

We discuss various properties for the Borel-Laplace sums w tri, j .

For any j ∈ Z and I j = I 0 + jπ =]0, π| + jπ, one can of course consider the Borel-Laplace sum w tri, j = S I j w, which defines a holomorphic function on the domain

• D(Ij, τ), a sectorial neighbourhood of ∞ with aperture ( I j = ( I 0 -jπ, ( I j =] - 3 2 π, + 1 2 π[-jπ.
Morever, for every j ∈ Z,

w tri, j+2 (z) = w tri, j (z) for z ∈ • D(Ij, τ) (3.32) because w ∈ C[[z -1 ]] 1 .
We mentioned in proposition 3.1 that the formal series w(z) is even. One deduces that for any θ ∈]0, π[, for every z ∈

• Π π-θ τ S π-θ w(z) = S -θ w(-z).
Therefore, for every j ∈ Z,

for every z ∈ • D(Ij, τ), w tri, j (z) = w tri, j+1 (-z). (3.33)
We know by proposition 3.1 that w(z) belongs to R[[z -1 ]]. This has the following consequence : for any

θ ∈]0, π[, for z ∈ • Π θ τ , S θ w(z) = S -θ w(z)
(where a stands for the complex conjugate of a ∈ C). In other words, for any j ∈ Z, the two functions w tri, j and w tri, j+1 are complex conjugate, for every z ∈ • D(Ij, τ), w tri, j (z) = w tri, j+1 (z).

(3.34)

However, neither w tri,0 nor w tri,1 are real analytic functions, since this would mean that the 1-coboundary w tri,0w tri,1 is zero which is not as we shall see later on. The properties (3.33) and (3.34) have the following consequences: for every j ∈ Z, w tri, j is " PT -symmetric" [10,11,13], in the sense that for every

z ∈ • D(Ij, τ), w tri, j (z) = w tri, j (-z). (3.35)
In particular, for r > 0 large enough, Along this state of mind, there are many ways of computing Borel-Laplace sums approximately in practice (see, e.g., [14,3]). Among them, one may quote the socalled hyperasymptotic methods [1] which have strong links with resurgence theory. These methods, originally arising from (and extending to) geometrical considerations on (multiple) singular integrals [23,9,8], can be applied to a wide class of problems stemming from applied mathematics and physics, see [20,21,22] and references therein. Other ways are available, for instance those based on the use of conformal mappings [2] with realistic upper bounds. It is also theoretically possible to calculate a 1-sum exactly by means of factorial series expansions [18,12].

w tri,0 (re -iπ/2 ) ∈ R, w tri,0 (re -iπ/2 ) ∈ iR. ( 3

Tritruncated solutions

Tritruncated solutions

One can easily translate proposition 3.14 into properties for the first Painlevé equation (2.1). However, to use the Boutroux's transformations (2.6), (2.7) properly, it is worth to work on the Riemann surface of the logarithm and we thus fix some notations.

Definition 3.18. We denote by C the Riemann surface of the logarithm,

C = {z = re iθ | r > 0, θ ∈ R}, π : z ∈ C → • z = re iθ ∈ C .
For any z = re iθ ∈ C, we refer to θ as to its argument, denoted by θ = arg z.

We denote by S 1 (usually identified with R) the set of directions of half-lines about 0 on C. We (still) denote by π : S 1 → S 1 the natural projection which makes S 1 an étalé space on S 1 (and even a universal covering).

Definition 3.19. Let θ ∈ S 1 be a direction and τ ∈ R. We set

Π θ τ = {z = re iα ∈ C | α ∈ ( θ and π(z) ∈ • Π θ τ }.
Let I ⊂ S 1 be an open arc and γ : I → R be a locally bounded function. We set

D(I, γ) = θ ∈I Π θ γ(θ ) ⊂ C. One calls D(I, γ) a sectorial neighbourhood of infinity on C.
In order to define the transformations (2.6) and (2.7) safely, we introduce a biholomorphic mapping.

Definition 3.20. The biholomorphic mapping T is defined by:

C T → C, z → x = T (z) = 30
(I, γ) = T D(I, γ) , • (I, γ) = π S (I, γ) . (3.38)
We will consider the domains D(I j , τ), j ∈ Z, for I j = I 0 + jπ =]0, π| + jπ and

τ(θ ) = 4.22 | sin(θ )| . Notice that D(I j+1 , τ) = e -iπ D(I j , τ) for any j ∈ Z.
The domain (I j , τ) (see Fig. 3.3 and Fig. 3.4) is a sectorial neighbourhood of in-

finity of aperture K j =] - 11 5 π, - 3 5 π[- 4 5
jπ and we may notice that, for any j ∈ Z, (I j+1 , τ) = e -4iπ/5 S (I j , τ). In particular,

• (I j+5 , τ) = • (I j , τ)
. We now think of w tri, j = S I j w as a holomorphic function on D(I j , τ). By (3.33) and (3.35), these functions satisfy some relationships: for any j ∈ Z, for every z ∈ D(I j , τ),

w tri, j (z) = w tri, j+1 (ze -iπ ), (3.39)
w tri, j (z) = w tri, j (ze -(2 j+1)iπ ),

with the convention z = re -iα ∈ C for z = re iα ∈ C.
This gives sense without ambiguity to (3.4), (2.6) and (2.7), with the transform . The dash lines recall the sectors (2.9).

z ∈ D(I j , τ) ↔ x ∈ (I j , τ) (3.40) w tri, j (z) ↔ u tri, j (x) = e iπ/2 √ 6 x 1/2 1 - 4 25 T -1 (x) 2 + w tri, j T -1 (x) T -1 (x) 2 .
The functions u tri, j are solutions of the first Painlevé equation (2.1) and by (3.39) and (3.40), they satisfy the following relationships: for any j ∈ Z, for every x ∈ (I j , τ),

u tri, j (x) = e 2iπ/5 u tri, j+1 (xe -4iπ/5 ), (3.41)
u tri, j (x) = e 2 5 (2 j+1)iπ u tri, j (xe -2 5 (4 j+7)iπ ),
We recover here the symmetries discussed in Sect. 2.5.

By projection, u tri, j becomes a function holomorphic on the domain

• (I j , τ)
. This provides five distinct functions u tri, j (x), j = 0, • • • , 4, the so-called tri-truncated solutions. We now use notions developed in [16] to which the reader is referred. Since w tri, j is a section on ( I j of A 1 , we deduce that the tritruncated solution u tri, j (x) belongs to the space of holomorphic functions with Gevrey asymptotic expansion of order 4/5 at infinity in • (I j , τ). One can thus recover u tri, j (x) by its asymptotics through 5/4-summability. It is also worth mentioning that u tri,2 (x) is a real analytic function, as a consequence of property (3.41). Proposition 3.15. Let be

• (I 0 , τ) = π T D(I 0 , τ) with τ(θ ) = 4.22 | sin(θ )|
and, u tri, j (x) = ω j u tri,0 ω -2 j x , ω j = e -2iπ 5 j , j = 0, • • • , 4, and u tri,2 is a real analytic function.

for j = 0, • • • , 4, • (I j , τ) = ω 2 j • (I 0 , τ), ω j = e -
Remark 3.4. It is shown in exercise 3.3 that for any j = 0, • • • , 4, the tri-truncated solution u tri, j can be analytically continued to the domain

• (I j , τ) with τ(θ ) = 1.4
| sin(θ )| . We will see later on that each tri-truncated solution u tri, j can be analytically continued to a wider domain than (3.42)

• (I j , τ).
P(∂ ) = n ∑ m=0 α n-m ∂ m ∈ C[∂ ], α 0 = 0, α n = 0 where G(z, y) is holomorphic in a neighbourhood of (z, y) = (∞, 0) ∈ C × C n , n ∈ N . We furthermore suppose that G(z, 0) = O(z -1 ) and ∂ |l| G(z, 0) ∂ y l = O(z -1 )
when |l| = 1.

1. Show that for every M ∈ N and up to making transformations of the type

w = M ∑ k=1 a k z -k + v, (3.43) one can instead assume that G(z, 0) = O(z -M-1 ). 2. We suppose that for some M ∈ N , G(z, y) satisfies G(z, 0) = O(z -M-1 ). Show
that, up to making a (so called) shearing transformation of the form

w = z -M v, (3.44)
one can rather assume that

G(z, 0) = O(z -1 ), ∂ |l| G(z, 0) ∂ y l = O(z -1 ) when |l| = 1 and ∂ |l| G(z, 0) ∂ y l = O(z -M(|l|-1) ) when |l| ≥ 2.
3. Deduce that, through transformations of the type (3.43) and (3.44), one can bring equation (3.42) under the prepared form:

P(∂ )w + 1 z Q(∂ )w = F(z, w, w , ..., w (n-1) ) (3.45)
P(∂ ) = n ∑ m=0 α n-m ∂ m ∈ C[∂ ] , Q(∂ ) = n-1 ∑ m=0 β n-m ∂ m ∈ C[∂ ]
where

F(z, y) is holomorphic in a neighbourhood of (z, y) = (∞, 0) ∈ C × C n such that F(z, 0) = O(z -2-M 0 ), M 0 ∈ N, ∂ |l| F(z, 0) ∂ y l = O(z -2 ) when |l| = 1 and
∂ |l| F(z, 0) ∂ y l = O(z -2-M |l| ), M |l| ∈ N, when |l| ≥ 2.

Show that the shearing transform

w = z -M v, M ∈ N , brings equation (3.45) into an equation of the form P(∂ )v + 1 z (Q(∂ ) -MP (∂ )) v = g(z, v, v , • • • , v (n-1) ).

3.2.

We consider the ODE (3.10) and its unique solution

w ∈ O( • R (0) ). 1. Show that, for any ρ ∈]0, 1[, for any ζ = ξ e iθ ∈ • R (0) ρ , ξ = |ζ |, ρ| w(ζ )| ≤ 392 625 + 7 ξ 0 | w(re iθ )| dr + 1 2 ξ 0 | w * 2 (re iθ )| dr.

We consider the ODE

y + y z -y = 392 625 z -4 + 1 2 y 2 . (3.46)
deduced from (3.2) by the transformation v(z) = 1 -4 25z 2 + y(z) or, from (3.6) through the transformation y(z) = z -2 w(z). In particular there exists a unique formal series y(z (3.46). We thus know that the formal Borel transform y belongs to M O( • R (0) ) and satisfies the convolution equation associated with (3.46) by formal Borel transformation:

) = z -2 w(z) ∈ C[[z -1 ]] solution of
(ζ 2 -1) y -1 * (ζ y) = 392 625 ζ 3 Γ (4) + 1 2 y * y. (3.47) 1. Let f ∈ O 0 be a germ such that f (0) = 0. Show that the solutions g ∈ O 0 of the convolution equation (ζ 2 -1)g -1 * (ζ g) = f are given by g(ζ ) = C (1 -ζ 2 ) 1/2 - f (ζ ) 1 -ζ 2 + 1 (1 -ζ 2 ) 1/2 ζ 0 η (1 -η 2 ) 3/2 f (η) dη, C ∈ C. (Hint : set g(ζ ) = G(ζ )
1-ζ 2 , differentiate the convolution equation to obtain a nonhomogeneous linear differential equation of order 1, and solve this equation).

Show that y satisfies the convolution equation (3.47) in M O(

• R (0) ) if and only if y satisfies the following fixed-point problem:

y = P 392 625 ζ 3 Γ (4) + 1 2 P y * y with Pg (ζ ) = - g(ζ ) 1 -ζ 2 + 1 (1 -ζ 2 ) 1/2 ζ 0 η (1 -η 2 ) 3/2 g(η) dη, (3.48) 3. Show that for any ρ ∈]0, 1[ and any ζ ∈ • R (0) ρ , and ζ (1-ζ 2 ) 3/2 ≤ 1 ρ 3/2 . 4. Show that for any ρ ∈]0, 1[ and any ζ ∈ • R (0) ρ , | y(ζ )| ≤ Y (ξ ) with ξ = |ζ |,
where Y is an entire function which solves the fixed-point problem:

Y = Q 392 625 ξ 3 Γ (4) + 1 2 Q Y * Y (3.49) QG (ξ ) = G(ξ ) ρ + 1 ρ 2 1 * G (ξ ) 5. For any ρ ∈]0, 1[ we denote by Y (z) the inverse Borel transform of Y . Show that Y (z) satisfies the algebraic equation ρ Y = 392 625 1 z 4 + 1 2 Y 2 1 + 1 ρz , Y (z) = 392 625 1 ρz 4 + O(z -5 ). (3.50) 6. We set U = D(∞, ρ 1.4 ). Show that the fixed-point problem (3.50) has a unique solution in B ρ 3 /2 = {H ∈ O(U), H ≤ ρ 3 2 }, for

3.4.

We consider the domain

• D(]0, π[, τ) for τ(θ ) = λ sin(θ ) , λ > 0.
We want to describe the boundary ∂

• D(]0, π[, τ) of this domain. 1. show that ∂ • D(]0, π[, τ
) is the envelope of the following family of line curves: Chapter 4

z = x + iy, x cos(θ ) -y sin(θ ) = λ sin(θ ) , θ ∈]0, π[. 2. Deduce that ∂ • D(]0, π[, τ) is the parabolic curve of equation y = x 2 4λ -λ . References 1. M.V.
A step beyond Borel-Laplace summability Abstract We previously showed that the minor w of the unique formal series solution w of the prepared ODE associated with the first Painlevé equation, defines a function holomorphic on a cut plane. We further analyze the analytic properties of w. We show in Sect. 4.5 how w can be analytically continued to a domain of a Riemann surface, defined in Sect. 4.2, and we draw some consequences. This question is related to the problem of mastering the analytic continuations of convolution products and, as a byproduct, of getting qualitative estimates on any compact set. This is what we will partly do in Sect. 4.3 and Sect. 4.4, using only elementary geometrical arguments. We end with some supplements in Sect. 4.6.

Introduction

We previously analyzed the Borel-Laplace summability of w(z) ∈ C[[z -1 ]], the unique formal solution of the prepared ODE (3.6) associated with the first Painlevé equation. This was done by two approaches. In one of them, we defined a sequence ( h l ) l∈N of Z-resurgent functions (proposition 3.8) and we showed that the minor w of w can be represented as the sum of the series ∑ l≥0 h l which converges to a holomorphic function on the cut plane

• R (0) = C \ {±[1, +∞[}.
The key issue is:

Question 4.1. Does w belong to the space of Z-resurgent functions or, in other words, is w a Z-resurgent formal series ?

The answer is "yes" and this will allow an in-depth examination of the (so-called) non-linear Stokes phenomenon for the first Painlevé equation, in the spirit of the various examples handled in [10]. However, this question requires further tools and we postpone the complete answer to the last chapter of this course. One of these tools consists in sharpening our understanding of Ω -resurgent functions, at least when Ω = Z. This is our aim in this chapter.

The Ω -resurgent functions have been recalled in definition 3.7. This can be rephrased as follows for Ω = Z: the germ ϕ ∈ O 0 is a Z-resurgent function if and only if ϕ can be represented by a function Φ holomorphic on U 0 = D(0, r), 0 < r ≤ 1, and for any given

ζ 0 ∈ U 0 = U \ {0}, this function can be analytically continued along any path γ of C \ Z originating from ζ 0 .
Notice in this rephrasing that U 0 could have been replaced by any connected and simply connected neighbourdhood of the origin, for instance • R (0) . (Exercise : why ?)

We would like to characterize Z-resurgent functions by means of Riemann surfaces. Let ϕ ∈ O 0 be a germ of holomorphic functions at 0 and (O, q) be the étalé space associated with the sheaf O (cf. remark 1.1). We denote by R( ϕ) the connected component of O containing ϕ. Endowed with the restricted projection q = q| R( ϕ) , R( ϕ) is the Riemann surface of ϕ.

We recall that a Riemann surface is a connected one-dimensional complex manifold [10,7,3]. Notice that R( ϕ) is not necessarily simply connected. (Exercise : why ?)

We now assume that ϕ is a Z-resurgent, determined by a function Φ holomorphic on U 0 ⊂ C, a connected and simply connected neighbourdhood of the origin. Let us draw some conclusions about R( ϕ) from this hypothesis. In the first place by the very construction of R( ϕ), one can find a neighbourhood U 0 ⊂ R( ϕ) of ϕ such that q (U 0 ) = U and the mapping q | U 0 : U 0 → U 0 is a homeomorphism. In particular, U 0 is connected and simply connected. In addition, let be ζ 0 ∈ U and denote by ϕ 0 = q | -1 U 0 (ζ 0 ) ∈ U 0 the germ of holomorphic functions at ζ 0 determined by Φ. Since ϕ is Z-resurgent, ϕ 0 can be analytically continued along any path γ of C \ Z originating from ζ 0 . In other words, any such path γ can be lifted to R( ϕ) from ϕ 0 with respect to q , and this lifting is unique by uniqueness of lifting [7]. We denote by Γ this lifting,

γ = q • Γ . Now assume that γ is a loop homotopic in C \ Z to a loop γ in U 0 . Then cont γ ϕ 0 = ϕ 0 because cont γ only depends on the homotopy class of γ in C \ Z, meanwhile ϕ 0 is represented by Φ ∈ O(U 0 ) on U 0 .
In regard, lifting the homotopy, Γ is homotopic to a loop in U 0 , thus null-homotopic since U 0 is simply connected.

This being said, we raise the following question:

Question 4.2. Can we determine a simply connected Riemann surface R Z on which any Z-resurgent function can be analytically continued ?

We answer to this question in Sect. 4.2, through an explicit construction of R Z . We also describe there various sheets of this Riemann surface which will be usefull for later purposes.

Next we turn to the convolution product. We already know by theorem 3.2 that the space of Z-resurgent functions is stable under convolution product. In other words, if the germs ϕ, ψ ∈ O 0 can be analytically continued to the Riemann surface R Z , then it is the same for their convolution product ϕ * ψ. But what about the question of upper bounds ? In the previous chapter, the answer was essentially the matter of lemma 3.3 and the new issue is: Question 4.3. Can we formulate an analogue of lemma 3.3 for holomorphic functions defined on the Riemann surface R Z ?

The main result of this chapter, namely theorem 4.1 and its corollaries detailed in Sect. 4.4, gives a partial to this question. Its proof relies on the use of shortest symmetrically contractile paths which we describe in Sect. 4.3. We then apply our results to the first Painlevé equation in Sect. 4.5, to get theorem 4.2. A theoretical supplement ends this chapter.

Resurgent functions and Riemann surface

This section is devoted to defining the Riemann surface R = R Z and some of its sheets. We first recall usual notations.

Notations

In this course, a path (or a parametrized curve) λ in a topological space X is any continuous function

λ : [a, a + l] → X, where [a, a + l] ⊂ R is a (compact) interval possibly reduced to {a}. One denotes by λ -1 the inverse path, that is λ -1 : t ∈ [a, a + l] → λ (2a + l -t)
We often work with standard paths, that is paths defined on

[0, 1]. The path λ : t ∈ [0, 1] → λ (a + tl) is the standardized path of λ . For two paths λ 1 : [a, a + l] → X, λ 2 : [b, b + k] → X so that λ 1 (a + l) = λ 2 (b)
, one denotes by λ 1 λ 2 their product (or also concatenation),

λ 1 λ 2 : t ∈ [a, a + l + k] → λ 1 (t), t ∈ [a, a + l] λ 2 (t -a -l + b), t ∈ [a + l, a + l + k]
We denote by ∼ X the equivalence relation of homotopy of paths with fixed extremities in X : λ 1 ∼ X λ 2 if the two paths λ 1 , λ 2 in X have same extremities and there exists a continuous map H : [0, 1] × [0, 1] → X that realizes a homotopy between the standardized paths λ 1 and λ 2 . When X has a (finite R-dimensional and C ∞ ) differential structure, one can define smooth paths. We recall that any path can be uniformaly approached by C ∞paths. Typically in this course, X = C with its 2-dimensional real differential structure. For a piecewise C 1 -path λ : I → C, its length is denoted by length(λ ) where

length(λ ) = n ∑ k=1 t k t k-1 |λ (t)|dt, for any partition 0 = t 0 < t 1 < • • • < t n = 1 of [0, 1]
for which λ has a continuous derivative on each interval

[t k-1 ,t k ].

4.2.2

The Riemann surface of Z-resurgent functions For any two

γ 1 , γ 2 ∈ R ζ 0 , γ 1 ζ 0
∼ γ 2 when one of the following conditions is satisfied:

• either γ 1 ∼ U 0 γ 2 or γ 1 ∼ C\Z γ 2
• or else there exists Notice that p ζ 0 is not a covering map since the curve lifting property [7,3] is not satisfied. For instance, as a rule, a path starting from and ending at 0 cannot be lifted from 0 on R Z,ζ 0 with respect to p ζ 0 . We precise the "pull back" of the complex structure.

γ 3 ∈ A ζ 0 ∩ B ζ 0 such that γ 1 ∼ U 0 γ 3 γ 2 ∼ C\Z γ 3 or γ 1 ∼ C\Z γ 3 γ 2 ∼ U 0 γ 3 . Exercise 4.1. Show that ζ 0 ∼ is an equivalence relation on R ζ 0 .
R Z,ζ 0 = {cl ζ 0 (γ) | γ ∈ R ζ 0 } and p ζ 0 : cl ζ 0 (γ) → γ(1) ∈ • R. ( 4
• Assume that ζ = cl ζ 0 (γ) with γ ∈ A ζ 0 (thus ζ ∈ U 0 ). Let U ⊂ U 0 be any connected and simply connected open neighbourhood of ζ . To (U, ζ ) we associate the set U ⊂ R Z,ζ 0 made of all ξ = cl ζ 0 (γ 1 γ 2 ) where γ 1 satisfies γ 1 ∼ U 0 γ while γ 2 is any path in U originating from ζ . • Assume that ζ = cl ζ 0 (γ) with γ ∈ B ζ 0 (in particular ζ = 0). Let U ⊂ C \ Z be
If U 1 , U 2 , U 1 ∩ U 2 = / 0 are two open sets of R Z,ζ 0 such that the mappings p ζ 0 | U 1 : U 1 → p ζ 0 (U 1 ) and p ζ 0 | U 2 : U 2 → p ζ 0 (U 2 ) are two homeomorphisms, then the chart transition p ζ 0 | U 2 • p ζ 0 | -1 U 1 : p ζ 0 (U 1 ∩ U 2 ) → p ζ 0 (U 1 ∩ U 2 ) is nothing but the identity map, thus is biholomorphic. This makes R Z,ζ 0 a Riemann surface. Exercise 4.3. Let U 0 , ζ 0 be as in definition 4.1. Let U 1 ⊂ C \ Z be a connected and simply connected open neighbourhood of ζ 0 such that U 0 ∩ U 1 is connected. We denote by U 0 ⊂ R Z,ζ 0 the uniquely defined open set such that p ζ 0 | U 0 : U 0 → U 0 is a homeomorphism and we set ζ 0 = p ζ 0 | -1 U 0 (ζ 0 ). We denote by U 1 ⊂ R Z,ζ 0 the uniquely defined neighbourdhood of ζ 0 such that p ζ 0 | U 1 : U 1 → U 1 is a homeomor- phism. 1. Show that U = U 0 ∪U 1 is simply connected. 2. We set U = U 0 ∪ U 1 . Show that p ζ 0 | U is a homeomorphism between U and U.

The Riemann surface R Z

Up to now, the Riemann surface (R Z,ζ 0 , p ζ 0 ) depends on the given of U 0 , a connected and simply connected neighbourdhood of the origin, and of ζ 0 ∈ U 0 . Lemma 4.1. Let U 0 (resp. U 1 ) be a connected and simply connected neighbourdhood of the origin in C and ζ 0 ∈ U 0 (resp. ζ 1 ∈ U 1 ). Then there exists a fiber preserving homeomorphism τ : R Z,

ζ 0 → R Z,ζ 1 between the Riemann surfaces (R Z,ζ 0 , p ζ 0 , 0) and (R Z,ζ 1 , p ζ 1 , 0).
Proof. Left as an exercise to the reader. Proposition 4.2. Let ϕ 0 ∈ O 0 be a germ of holomorphic functions at the origin and let (R( ϕ 0 ), q, ϕ 0 ) be its Riemann surface. Then ϕ 0 is a Z-resurgent function if and only if (R, p, 0) is contained in (R( ϕ 0 ), q, ϕ 0 ), that is there exists a fiber preserving continuous map τ : R → R( ϕ 0 ), q • τ = p and τ(0) = ϕ 0 .

Proof. Assume that ϕ 0 is a Z-resurgent function. We set U 0 = D(0, 1) and we pick a point ζ 0 ∈ U 0 . On the one hand, there is a uniquely determined domain U 0 ⊂ R homeomorphic to U 0 by p| U 0 and we set ζ 0 = p| -1 U 0 (ζ 0 ). On the other hand, there is a uniquely determined domain U 0 ⊂ R( ϕ 0 ) homeomorphic to U 0 by q| U 0 and

we set ζ 0 = q| -1 U 0 (ζ 0 ).
We get this way a natural fiber preserving homeomorphism

τ| U 0 : ζ ∈ U 0 → ζ ∈ U 0 .
We now extend τ| U 0 as follows: pick any path γ in C \ Z, originating from ζ 0 , let Γ be its lifting from ζ 0 on R with respect to p and set

ζ = Γ (1) ∈ R.
The path γ can be lifted as well on R( ϕ 0 ) with respect to q from ζ 0 into a path Γ , because ϕ 0 is Z-resurgent. We set Γ (1) = ζ . The extended mapping τ : ζ ∈ R → R( ϕ 0 ) thus (well)-defined is injective by uniqueness of lifting [7], continuous because we work with étalé spaces, and preserves fibers. The converse of the proposition is left to the reader as an exercise.

In other words, ϕ 0 ∈ O 0 is a Z-resurgent function if and only ϕ 0 can be analytically continued to the Riemann surface R Z . This means that one can identify the space R with the space O(R) of functions holomorphic on the Riemann surface R. Definition 4.5. The Riemann surface (R Z , p, 0) is called the Riemann surface of Z-resurgent functions.

Riemann surface R Z and sheets

We introduce various sheets and domains on R Z . At first sight artificially complicated, these constructions will be needed to state one of main results of this chapter, namely theorem 4.1 and its consequences.

Principal sheet

By the very construction of the Riemann surface R, there exists a unique domain R (0) of R so that p| R (0) realizes a homeomorphism between R (0) and the simply connected domain

• R (0) . The domain R (0) is made of endpoints ζ of paths deduced from any segment [0, ζ ] ⊂ • R (0)
, by lifting from 0 with respect to p. Definition 4.6. One refers to R (0) as to the principal sheet of the pointed Riemann surface (R, 0). For every ρ ∈]0, 1[, one denotes by R (0)

ρ the unique open subset of R (0) such that p(R (0) ρ ) = • R (0) ρ . (See Fig. 4.1). 4.2.3.2 Other sheets Definition 4.7. Let be m ∈ N , ε = (ε 1 , • • • , ε m-1 ) ∈ {+, -} m-1 a (m -1)-tuple of signs and n = (n 1 , • • • , n m-1 ) ∈ (N ) m-1 a (m -1)
-tuple of positive integers. Let θ 1 ∈ {0, π} ⊂ S 1 be a direction. Let γ be a path in C originating from 0. When m = 1, one says that the path γ is of type γ θ 1 () when γ closely follows the segment e iθ 1 ]0,

1[=]0, ω 1 [ toward ω 1 = e iθ 1 .
Otherwise, for m ≥ 2, on says that the γ is of type type

γ θ 1 ε n if γ connects the segment ]0, ω 1 [ to the segment ]ω m-1 , ω m [, ω m -ω m-1 = e iθ m ,
through the following steps:

• γ closely follows the segment ]0, ω 1 [ toward the direction θ 1 , makes n 1 half-turns around the point ω 1 , anti clockwise when ε 1 = +, clockwise when ε 1 = -1, and finally closely follows the segment ]ω 1 , ω 2 [, ω 2ω 1 = e iθ 2 , toward the direction for θ = 0, ε = (-, +, +, +, -) and n = (1, 2, 1, 1, 1).

θ 2 = θ 1 + ε 1 (n 1 -1)π; • then, successively for k = 2, • • • , m -1, γ makes n k half-turns around the point ω k , anti clockwise when ε k = +, clockwise when ε k = -1, and eventually closely follows the segment ]ω k , ω k+1 [, ω k+1 -ω k = e iθ k+1 , toward the direction θ k+1 = θ k + ε k (n k -1)π. -1 -2 -3 1 0 2 0 -1 -2 -3 1 2 Fig. 4.1 Above, the domain • R (0) . Below, the domain • R (0) ρ .
0 1 2 -1 -2 When n = (1, • • • , 1) ∈ {1} m-1
, we simply say that γ is of type γ θ 1 ε . (See Fig. 4.2 and Fig. 4.3).

For instance, if γ is of type γ θ ε , then someone standing at 0 ∈ C and looking in the direction of the half-line ]0, e iθ ∞[ will see the path γ avoiding the point ω n = ne iθ ∈ C by swerving in the direction of his right hand when ε n = +, of his left hand when ε n = -.

Definition 4.8. Let be m ∈ N , ε ∈ {+, -} m , n ∈ (N ) m and θ ∈ {0, π}. We denote by R ε n ,θ ⊂ R the domain made of endpoints ζ = Γ (1)
where Γ is the lift from 0 with respect to p of any path γ of the form γ = γ 1 γ 2 with the conditions : γ 1 is a path of type γ θ ε n ending at

• ξ ∈]p, (p + 1)[=]ω m , ω m+1 [, γ 2 is a path starting from • ξ , and contained in the simply connected domain C \ {] -∞, p] ∪ [p + 1, +∞[}, star-shaped from • ξ . When n = (1, • • • , 1) ∈ {1} m , we simply write R ε,θ = R ε n ,θ .
The collection of sheets {R (0) , R ε n ,θ } provides an open covering of R, with the following property: the restriction p| R ε n ,θ is a homeomorphism between R ε n ,θ and the simply connected domain

C \ {] -∞, p] ∪ [p + 1, +∞[} where ]p, (p + 1)[=]ω m , ω m+1 [,
with ω m , ω m+1 as given by definition 4.7.

Remark that for every θ ∈ {0, π}, for every m ∈ N and for every ε either in {+} m or in {-} m , R (0) and R ε,θ have a non-empty intersection (a half-plane on projection). This justifies the following definitions. Definition 4.9. Let be m ∈ N . We set (+) m-1 = (+,

• • • , +) ∈ {+} m-1 and (-) m-1 = (-, • • • , -) ∈ {-} m-1 . We denote by (±) m-1 any (m -1)-tuple of the form (±, • • • , ±) ∈ {+, -} m-1 . Also, (+) 0 = (-) 0 = (±) 0 = () is the 0-tuple. Thus the set of all (±) m is made of 2 m elements. Definition 4.10. The domain R ε,θ is called a R (0) -nearby sheet if ε ∈ m∈N {(+) m , (-) m }.
One denotes by R (1) ⊂ R the union of the principal sheet and of all nearby sheets: R (1)

= R (0) θ ∈{0,π}, m∈N R (+) m ,θ ∪ R (-) m ,θ .
More generally, for any k ∈ N , one defines:

R (k+1) = R (k) θ ∈{0,π},m∈N n∈(N ) k R ((±) n k ,(+) m-1 ),θ ∪ R ((±) n k ,(-) m-1 ),θ . Remark 4.1. Notice that p(R (+) m ,θ ) = p(R (-) m ,θ ) = C \ e iθ {] -∞, m] ∪ [m + 1, +∞[} and k R (k) = R.
For every integer k ∈ N, the domain R (k) inherits from R the structure of complex manifold, thus is is a Riemann surface.

0 1 2 -1 -2 -3 Fig. 4.4 The domain • R ρ when 1 9 < ρ ≤ 1 7
(the scale is not correct).

R (0)

ρ -nearby domains

Our aim is to introduce various of the Riemann surface R which will be convenient for later purposes.

We start with the following remark: for ρ ∈]0, 1[ and m ∈ N , the closed discs D(m, mρ) and D(m + 1, (m + 1)ρ) are disjoint as soon as m < ρ -1 -1 2 . Thus, now assuming that ρ ∈]0, 1 5 [ and introducing the integer part

M (ρ) + 1 = ρ -1 -1 2 ≥ 2 ( . ]
E m,θ ρ = (ζ ,ξ )∈D e iθ m ×D e iθ (m+1) ξ + t(ξ -ζ ), ζ + t(ζ -ξ ) | t ∈ [0, +∞[ and • Q m,θ ρ = C\E θ ρ,m . for any integer m > M (ρ), we set • Q m,θ ρ = / 0 . For any positive integer m ≥ 1 and ε = ±, we set • Q (ε) m ,θ ρ = • Q m,θ ρ ∩{ζ | εe iθ (ℑζ ) ≤ 0}. See Fig. 4.5.
The domains

• Q m,θ
ρ have been defined so as to enjoy the following property :

Lemma 4.3. Let be ζ ∈ • Q m,θ
ρ for some integer m ∈ [1, M (ρ)] and some θ ∈ {0, π}. Then, for every integer n ∈

[1, m], ζ -D e iθ n is a subset of • Q m-n,θ ρ . -3 2 -2 -1 0 1 Fig. 4.5 The domain • Q 2,π ρ . The set • Q (-) 2 ,π ρ
lies below the real axis,

• Q (+) 2 ,π ρ
lies above the real axis.

Proof. We only consider the case θ = 0 and we suppose ζ

∈ • Q m,0 ρ . Le be n ∈ [1, m]. Assume the existence of ζ n ∈ D n such that ζ -ζ n / ∈ • Q m-n,0 ρ , thus ζ -ζ n ∈ E m-n,0 ρ (see definition 4.12). Therefore, there exist ζ m-n ∈ D m-n , ζ m-n+1 ∈ D m-n+1 and t ∈ [0, +∞[ such that ζ -ζ n = ζ m-n + t(ζ m-n -ζ m-n+1 ) or ζ -ζ n = ζ m-n+1 + t(ζ m-n+1 -ζ m-n ).
We look only at the first case, which we write as follows:

ζ = (ζ m-n + ζ n ) + t (ζ m-n + ζ n ) -(ζ m-n+1 + ζ n ) . We observe that ζ m-n + ζ n ∈ D m while ζ m-n+1 + ζ n ∈ D m+1 . Therefore ζ ∈ E m,0
D m,θ ρ = tζ | t ∈] -∞, 1], ζ ∈ D e iθ m ∪ tζ | t ∈ [1, +∞[, ζ ∈ D e iθ (m+1) .
We set

• P m,θ ρ = C \ D m,θ
• R (ε) m ,θ ρ the domain • R (ε) m ,θ ρ = • P (ε) m ,θ ρ ∪ • Q (-ε) m ,θ ρ (see Fig. 4.7
), and we set:

• R m,θ = 0<ρ≤1/5 • R (+) m ,θ ρ = 0<ρ≤1/5 • R (-) m ,θ ρ = C \ e iθ ] -∞, m] ∪ [m + 1, +∞[ .
+ λ - -1 -2 -3 1 0 2 -3 2 -1 0 1 -2 Fig. 4.7 Figure above, the domain • R (+) 2 ,π ρ . Figure below, the domain • R 2,π .
We have already noticed that for θ ∈ {0, π} and m ∈ N , the restriction p| R (+)m,θ and p| R (-)m,θ respectively, realises a homeomorphism between the nearby sheet R (+) m ,θ and R (-) m ,θ respectively, and the simply connected domain

p(R (+) m ,θ ) = p(R (-) m ,θ ) = • R m,θ .
This justifies the following definition.

R (ε) m ,θ ρ = p| -1 R (ε)m,θ • R (ε) m ,θ ρ . The domain R (ε) m ,θ ρ is called a R (0)
ρnearby domains .The connected and simply connected domain R

(1)

ρ ⊂ R (1) is de- fined by R (1) ρ = R (0) ρ 1≤m≤M (ρ) θ ∈{0,π},ε=± R (ε) m ,θ ρ
. We denote by R

ρ the closure of R

(1) ρ in R (1) .

Observe that p R

Geodesics

The closed space

in R (+) 3 ,0 ρ \ R (0) ρ . 0 ζ 1 2 3 D D D
In this lemma, a geodesic means a locally shortest path for the euclidean metric. Lemma 4.5 can be seen as a corollary of the Hopf-Rinow theorem [9]. As a matter of fact, the situation is quite simple here : inside X, a geodesic is nothing but a straight line, otherwise one just follows the smooth boundary ∂ X. (See [1] and references therein for more general cases.)

The Riemann surface (R, p, 0) of Z-resurgent functions can also be thought of as a real 2-dimensional Riemannian manifold, by pulling-back by p the standard euclidean metric on the complex plane. It follows from its very construction that R

ρ ⊂ R is a complete real 2-dimensional Riemannian manifold. Pick a point ζ ∈ R (1)
ρ . Up to homotopy, there exists a unique path Λ joining 0 to

ζ in R (1) ρ , because R (1)
ρ is (path)connected and simply connected. Moreover, from the Hopf-Rinow theorem, Λ can be chosen as a shortest (C 1 -)path in this homotopy class, and is uniquely determined when parametrized by arc-length. To sum up: Second case:

ζ belongs to R (ε) m ,θ ρ \R (0) ρ for some θ ∈ {0, π}, ε = ± and some m ∈ [1, M (ρ)].
Consider the path λ = γ 0 δ 0 δ 1 where γ 0 , δ 0 , δ 1 stands for the following geodesics with their arc-length parametrizations (see Fig. 4.8) : [1, m] to the right when ε = + and to the left when ε = -;

• γ 0 follows the segment [0, • ζ 0 ] ⊂ ∂ • P (ε) m ,θ ρ ∩ • R (0) ρ that circumvents the segment e iθ
• δ 0 is the arc-curve from • ζ 0 to • ζ 1 that follows in • R (ε) m ,θ ρ the boundary ∂ D e iθ m ; • δ 1 follows the segment [ • ζ 1 , • ζ ] in • R (ε) m ,θ ρ (possibly reduced to the point • ζ ).
Once again, one deduces Λ from λ by lifting. given by lemma 4.7 is called the shortest path from 0 to ζ .

1. H t (0) = 0 and H t can be lifted on the Riemann surface R Z with respect to p from 0; 2. H t (1) -H t (s) = H -1 t (s) for every s ∈ I. The path H 0 (resp. H 1 ) is called the initial path of H (resp. final path) and the path t ∈ I → H t (1) is called the endpoint path of H. A path λ in C is called a symmetrically Z-contractile path if its standardized path λ is the final path of a symmetric Z-homotopy whose initial path follows a segment

[0, ζ ] of • R in the forward direction.
Let ζ be any point of the Riemann surface R Z and pick a path joining 0 to ζ , thus uniquely defined up to homotopy. It is known that one can find a path Λ in this homotopy class with the further condition : its projection λ = p • Λ is a symmetrically Z-contractile path. This is a key result to analyze the convolution product, as detailed in [10].

However, there are plenty of paths with the above properties and we raise the question: Question 4.4. In the homotopy class of these paths, is it possible to find a shortest curve ?

This question is meaningless because R Z is not a complete Riemannian manifold, but makes sense on R ρ originating from 0 and let λ = p • Λ be its projection. The path Λ is said to be symmetric if λ satisfies the condition:

λ (1) -λ (s) = λ -1 (s)for every s ∈ [0, 1]. A symmetric path Λ in R (1)
ρ is said to be shortest-symmetric when Λ is a shortest (C 1 -)path among the symmetric paths belonging to the same homotopy class in R

ρ and Λ be the shortest path from 0 to ζ . We denote by λ = p • Λ its projection.

First case: Either ζ belongs to R (0) ρ . Then λ follows the segment [0, • ζ ] ⊂ • R (0) ρ . We set H : (s,t) ∈ I × I → H(s,t) = s • ζ ∈ • R (0) ρ . For each t ∈ I, the path H t : s ∈ I → H(s,t) can be lifted uniquely on R (0) ρ from 0 with respect to p into a path H t : I → R (0)
ρ . From the lifting theorem for homotopies [7,3], the mapping H : (s,t) ∈ I × I → H t (s) is continuous and matches the other conditions.

second case: Or else ζ belongs to R (ε) m ,θ ρ \ R (0) ρ for some θ ∈ {0, π}, ε = ± and some m ∈ [1, M (ρ)].
For simplicity, we suppose θ = 0 and ε = +. The path λ , resp. Λ , can be written as a product λ = γ 0 λ 1 , resp. Λ = Γ 0 Λ 1 , where γ 0 = p •Γ 0 and λ 1 = δ 0 δ 1 = p • Λ 1 are the geodesics described in remark 4.2 with their arc-length parametrizations.

We set H 0 = γ 0 , resp. H 0 = Γ 0 the standardized path deduced from γ 0 , resp. Γ 0 , which follows a segment in

• R (0) ρ , resp. R (0)
ρ . This path can be lifted from 0 with respect to p into a unique path H 0 whose endpoint is denoted by ζ 0 = H 0 (1). By its very construction, the point ζ 0 belongs to V , the closure of R

(+) m ,0 ρ \ R (0)
ρ , and we can apply lemma 4.4. Therefore H 0 can be thought of as a geodesic in

X ζ 0 = • R ρ \ 1≤n≤m { • ζ 0 -D n }
and is a shortest path in its homotopy class, by application of lemma 4.5.

According to lemma 4.4 again, the space

X ξ = • R ρ \ 1≤n≤m { • ξ -D n } remains in
the field of application of lemma 4.5 for every ξ ∈ V . One gets this way a local system X ξ ξ ∈V of Riemannian manifolds with smooth boundary. Let 1 > 0 be the length of Λ 1 and

T : t ∈ [0, 1] → t 1 ∈ [0, 1 ]. For any t ∈ [0, 1], the point ζ t = Λ 1 • T (t) belongs to V by construction. To the path Λ 1 • T is associated a section t ∈ [0, 1] → X ζ t , thus a map t ∈ [0, 1] → [γ t ]
which allows to follow the continuous deformation of the homotopy class [γ 0 ] of γ 0 , the extremities 0 and ζ 0 being kept fixed. In the homotopy class

[γ t ] we choose, for any t ∈ [0, 1], a shortest path γ t in X ζ t with its arc-length parametrization. Let K t = γ t λ 1 | [0,T -1 (t)] be the (minimal)
ζ in R (+) 3 ,0 ρ \ R (0) ρ 0 ζ 3 D 2 D 1 D 1 D 2 D 3 D ζ- ζ- ζ-
geodesic defined as the product of γ t with the restriction of λ 1 to [0, T -1 (t)]. We denote by H t the path deduced from K t by standardization and we eventually obtain a continuous mapping H : (s,t)

∈ I × I → H(s,t) = H t (s), I = [0, 1]. See Fig. 4.9.
For any t ∈ [0, 1], the path H t can be lifted to R

ρ from 0 with respect to p. This gives a path, denoted by H t , which is shortest symmetric by construction and the mapping H : (s,t)

∈ I × I → H (s,t) = H t (s) ∈ R (1)
ρ is continuous by the lifting theorem for homotopies [7,3]

ρ → leng ρ (ζ ) ∈ R + is continuous. Lemma 4.9. Let be ζ ∈ R (1)
ρ and H be its associated shortest symmetric (Z, ρ)homotopy. Then for every t ∈ [0, 1] and every s ∈ [0, 1]:

• leng ρ H t (s) ≤ length H t | [0,s] ; • leng ρ H -1 t (s) ≤ leng ρ H t (1) -length H t | [0,s] .
We finally state a result drawn from [8], which gives an upper bound for the length of the shortest symmetrically contractile path we work with.

| • ζ | ≤ leng ρ (ζ ) ≤ 1 ρ | • ζ | + 1 ρ 1 ρ -2 .
Proof. The first case is obvious. The second case means that

ζ ∈ R (ε) m ,θ ρ \ R (0) ρ for some θ ∈ {0, π}, ε = ± and some m ∈ [1, M (ρ)].
Let us assume that θ = 0 and ε = + for simplicity. We return to the construction of the shortest symmetrically contractile path H 1 associated with ζ (see also Fig. 4.9) and we denote by

H 1 = p • H 1 its projection.
The path H 1 is made of :

• m + 1 segments between ∂ D n and ∂ • ζ -D m-n , n ∈ [0, m]. Each of these seg- ments has length less than | • ζ -m| + mρ. • m segments between ∂ • ζ -D m-n and ∂ D n+1 , n ∈ [1, m]. Each of these segments has length less than | • ζ -(m + 1)| + (m + 1)ρ.
• 2m arcs of circle, the total length of which being less than 2(1

+ • • • + m)2πρ.
Putting things together:

leng ρ (ζ ) ≤ (2m + 1)| • ζ | + 2m(m + 1)(1 + ρ) + 2m(m + 1)πρ. Since ρ ≤ 1 5 , one has | • ζ | ≤ leng ρ (ζ ) ≤ (2m + 1)| • ζ | + 4m(m + 1). Remember that M (ρ) + 1 = ρ -1 -1 2 , thus m ≤ M (ρ) ≤ 1
2ρ -1 and one concludes.

Convolution product and related properties

It has been recalled that the space R = RZ of Z-resurgent germs is a convolution algebra without unit (definition 3.7 and theorem 3.2).

Question 4.5. Is it possible to give quantitave estimates for the convolution product of two Z-resurgent functions ?

The answer is "yes", as detailed (without proof) in [10] (see also [2,12]). Even, quantitave estimates can be obtained for iterated convolutions and this allows nonlinear operations in the frame of resurgent functions [14].

Nevertheless, these results are difficult to use in our context. This is why we follow another strategy in the sequel.

A new convolution algebra

Definition 4.20. Let k ∈ N be a positive integer. We denote by R(k) the space of germs of holomorphic functions at the origin which can be analytically continued to the Riemann surface R (k) .

In other words, the germ ϕ ∈ O 0 belongs to R(k) when there exists a function

Φ ∈ O(R (k) ) holomorphic on R (k) whose germ φ 0 ∈ O 0 at 0 satisfies φ 0 = ϕ • p. Notice that the linear map ∂ : g ∈ R(k) → -ζ g still provides a derivation of R(k) .
Theorem 4.1. 1. The space R(1) is a convolution algebra (without unit). 2. Let ϕ, ψ ∈ R(1) be two germs and let Φ,Ψ ∈ O(R (1) ) be their associated holomorphic functions on R (1) . Assume that the following properties hold : for ev-

ery ζ ∈ R (1) ρ , Φ(ζ ) ≤ F leng ρ (ζ ) and Ψ (ζ ) ≤ G leng ρ (ζ )
, where F, G are two positive, non-decreasing and continuous functions on R + . Then the convolution product ϕ * ψ, resp. ( ∂ ϕ) * ψ, can be analytically continued to R (1) and the corresponding function χ ∈ O(R (1) ), resp. ϒ ∈ O(R (1) ), satisfies the following properties: for every

ζ ∈ R (1) ρ , χ(ζ ) ≤ F * G leng ρ (ζ ) , resp. ϒ (ζ ) ≤ leng ρ (ζ ) F * G leng ρ (ζ ) .
Proof. The standard proof for proving that R is a convolution algebra [10,12] can be copied as it stands for R (1) . We sketch it here, essentially so as to fix notations that will be used later on, more details can be found in [10]. Let be ϕ, ψ ∈ R(1) and let Φ,Ψ ∈ O(R (1) ) be their associated holomorphic functions on R (1) . The convolution product ϕ * ψ( 0) , thus there exists a uniquely determined point denoted by

• ζ ) is well-defined for every ζ ∈ R (0) and we set χ(ζ ) = ϕ * ψ( • ζ ). For every • ζ 0 ∈ • R (0) ρ and • ξ ∈ C such that | • ξ | < ρ 2 , the point • ζ 0 + • ξ belongs to • R (
ζ 0 + • ξ ∈ R (0) such that p(ζ 0 + • ξ ) = • ζ 0 + • ξ .
Therefore, the convolution product

χ(ζ 0 + • ξ ) = ϕ * ψ( • ζ 0 + • ξ ) reads : χ(ζ 0 + • ξ ) = • ζ 0 + • ξ 0 ϕ(η) ψ( • ζ 0 + • ξ -η)dη = • ζ 0 0 ϕ(η) ψ( • ζ 0 + • ξ -η)dη + • ξ 0 ϕ( • ζ 0 + η) ψ( • ξ -η)dη.
Let now ζ be any given point in R

ρ . We denote by H the associated shortest symmetric (Z, ρ)-homotopy given by proposition 4.3. We want to construct the analytic continuation of χ at ζ . We therefore assume that ζ 0 is the endpoint H 0 (1) of the intial path H 0 . The above equality yields :

χ H 0 (1) + • ξ = 1 0 Φ(H 0 (s))Ψ (H -1 0 (s) + • ξ )H 0 (s)ds + • ξ 1 0 Φ(H 0 (1) + • ξ s)ψ • ξ (1 -s) ds.
where H 0 = p • H 0 stands for the projection of H 0 . The analytic continuation of

χ from H 0 (1) along the path t ∈ [0, 1] → H t (1) ∈ R (1)
ρ is thus given by (see the arguments in [10]):

χ H t (1) + • ξ = 1 0 Φ(H t (s))Ψ (H -1 t (s) + • ξ )H t (s)ds + • ξ 1 0 Φ(H t (1) + • ξ s)ψ • ξ (1 -s) ds.
In particular when

ζ = H 1 (1), χ(ζ ) = 1 0 Φ H 1 (s) Ψ H -1 1 (s) H 1 (s)
ds where H 1 is the shortest symmetrically contractile path associated with ζ .

Notice that the germ χ(ζ + • ξ ) of holomorphic functions at ζ thus obtained does not depend on the chosen path H 1 since R (1) is simply connected.

We turn to estimates. Let T be the homothety s ∈ [0, 1] → s.leng ρ (ζ ) so that

χ(ζ ) = leng ρ (ζ ) 0 Φ H 1 • T -1 ( ) Ψ H -1 1 • T -1 ( ) d .
We then use lemma 4.9 to get:

|χ(ζ )| ≤ leng ρ (ζ ) 0 F G leng ρ (ζ ) -d ≤ F * G leng ρ (ζ ) .
The proof for the last assertion is left as an exercise.

Convolution space and uniform norm

The following definition makes sense by lemma 4.8 and lemma 4.9. Definition 4.21. Let L > 0 be a real positive number and ρ ∈]0, 1 5 ]. One denotes by

U ρ,L the open subset of R (1) ρ defined by: U ρ,L = {ζ ∈ R (1) ρ | leng ρ (ζ ) < L}. An element of U ρ,L is called a L-point.
We denote by O(U ρ,L ) the space of functions holomorphic on U ρ,L and continuous on U ρ,L . For any two elements f , g ∈ O(U ρ,L ), one sets

f * g(ζ ) = leng ρ (ζ ) 0 f H 1 • T -1 ζ ( ) g H -1 1 • T -1 ζ ( ) d (4.2)
where H 1 stands for the shortest symmetrically contractile path associated with

ζ ∈ U ρ,L , while T ζ is the homothety T ζ : s ∈ [0, 1] → s.leng ρ (ζ ). The function ζ ∈ U ρ,L → f * g(ζ )
is called the convolution product of f and g. Proof. Use lemma 4.9 and adapt the proof of theorem 4.1.

The following definition is similar to definition 3.12.

Definition 4.22. Let U = U ρ,L be an open set of L-points. We denote by M O(U ) the maximal ideal of O(U )) defined by M O(U ) = { f ∈ O(U ), f (0) = 0}.
Let ν ≥ 0 be a nonnegative real number. The norm . ν on O(U )) is defined by

f ν = L sup ζ ∈U e -νleng ρ (ζ ) f (ζ ) . We extend this norm to Cδ ⊕ O(U ρ,L ) by setting cδ + f ν = |c| + f ν for every f ∈ O(U ) and every c ∈ C.
We now state an analogue of proposition 3.9.

Proposition 4.5. The normed space Cδ ⊕ O(U ), . ν is a Banach algebra. In particular, for every f

, g ∈ Cδ ⊕ O(U , f * g ν ≤ f ν g ν . The space M O(U ) is closed in O(U ), . ν . Moreover, for ν > 0: 1. for every n ∈ N, for every g ∈ O(U ), (ζ → • ζ n ) * g ν ≤ n! ν n+1 g ν , (ζ → • ζ n ) ν ≤ n! ν n+1 L and (ζ → 1) ν = L. 2. for every f , g ∈ O(U ), f g ν ≤ 1 L f ν g 0 . 3. for every f ∈ O(U ), ν ≥ ν 0 ≥ 0 ⇒ f ν ≤ f ν 0 . 4. for every f ∈ M O(U ), lim ν→∞ f ν = 0. 5. the derivation ∂ |O(U ) : f ∈ O(U ) → - • ζ f ∈ M O(U ) is invertible and the in- verse map ∂ -1 satisfies: for every f ∈ O(U ), for every g ∈ M O(U ), ∂ -1 ( f * g) belongs to M O(U ) and ∂ -1 ( f * g) ν ≤ 1 νL f ν ∂ -1 g 0 . For every Cδ ⊕ O(U ), for every g ∈ M O(U ), ∂ -1 ( f * g) belongs to O(U ) and ∂ -1 ( f * g) ν ≤ f ν ∂ -1 g ν .
Proof. The norm . ν is obviously equivalent to the maximum norm on the vector space O(U ). This shows the completeness of O(U ), . ν and of

Cδ ⊕ O(U ), . ν as well. Pick a point ζ ∈ U . For any f , g ∈ O(U ), f * g(ζ ) = leng ρ (ζ ) 0 d e ν leng ρ H 1 •T -1 ζ (l) +leng ρ H -1 1 •T -1 ζ (l) (4.3) × f H 1 • T -1 ζ ( ) e -νleng ρ H 1 •T -1 ζ (l) g H -1 1 • T -1 ζ ( ) e -νleng ρ H -1 1 •T -1 ζ (l) .
We know from lemma 4.9 that leng ρ (H

1 (s)) + leng ρ (H -1 1 (s)) ≤ leng ρ (ζ ) for any s ∈ [0, 1]. Therefore Le -νleng ρ (ζ ) | f * g(ζ )| ≤ f ν g ν leng ρ (ζ ) 0 1 L d ≤ f ν g ν .
This shows that for every f ,

g ∈ O(U ), f * g ν ≤ f ν g ν , hence O(U ), . ν
is a Banach algebra and Cδ ⊕ O(U ), . ν as well. We encourage the reader to show the other properties.

Remark 4.4. We have already noticed that the space R(1) can be identified with the space O(R (1) ) of holomorphic functions on the Riemann surface R (1) . Since

O(R (1) ) = L>0 0<ρ≤1/5 O(U ρ,L ), formula (4.
2) provides the convolution product on O(R (1) ).

An extended Grönwall-like lemma

The following statement is similar to lemma 3.9.

Lemma 4.11 (Extended Grönwall lemma). Let N ∈ N be a positive integer. Let and ( f n ) 0≤n≤N , resp. ( F n ) 0≤n≤N , be a (N + 1)-tuple of functions in O(R (1) ), resp. of entire functions, real, positive and non-decreasing on R + , with at most exponential growth of order 1 at infinity. We suppose that for every 0 ≤ n ≤ N and every

ζ ∈ R (1) ρ , | f n (ζ )| ≤ F n leng ρ (ζ ) . Otherwise, let p, q, r be polynomial functions such that the function ζ → p(- • ζ ) does not vanish on R (1)
ρ and we assume that the following upper bounds are valid:

a = sup ζ ∈R (1) ρ |q|(leng ρ (ζ )) |p(- • ζ )| < ∞, b = sup ζ ∈R (1) ρ |r|(leng ρ (ζ )) |p(- • ζ )| < ∞, c = sup ζ ∈R (1) ρ 1 |p(- • ζ )| < ∞.
We furthermore assume that w ∈ O(R (1) ρ ) solves the following convolution equation:

p( ∂ ) w + 1 * [q( ∂ ) w] = ζ * [r( ∂ ) w] + f 0 + N ∑ n=1 f n * w * n . (4.4)
Then, for every d ≥ 0 and every ζ ∈ R

w = d + [a + bξ ] * w+c F 0 + N ∑ n=0 F n * w * n . (4.5) Proof. Let w ∈ O(R (1)
ρ ) be a solution of convolution equation (4.4). This means that for every

ζ ∈ R (1) ρ : p( ∂ ) w(ζ ) = f 0 (ζ ) - leng ρ (ζ ) 0 q H 1 • T -1 ζ ( ) w H 1 • T -1 ζ ( ) d + leng ρ (ζ ) 0 H -1 1 • T -1 ζ ( )r H 1 • T -1 ζ ( ) w H 1 • T -1 ζ ( ) d + N ∑ n=1 leng ρ (ζ ) 0 f n H -1 1 • T -1 ζ ( ) w * n H 1 • T -1 ζ ( ) d .
where H 1 stands for the shortest symmetrically contractile path associated with ζ . From lemma 4.9 and the hypotheses, one obtains with ξ = leng ρ (ζ ):

| w(ζ )| ≤ 1 |p(- • ζ )| F 0 (ξ ) + ξ 0   |q|( ) |p(- • ζ )| + |r|( ) |p(- • ζ )| (ξ -)   | w H 1 • T -1 ζ ( ) | d + N ∑ n=1 ξ 0 1 |p(- • ζ )| F n (ξ -)| w * n H 1 • T -1 ζ ( ) | d . Therefore | w(ζ )| ≤ c F 0 (ξ ) + ξ 0 [a + b(ξ -s)] | w H 1 • T -1 ζ ( ) | d (4.6) + c N ∑ n=1 ξ 0 F n (ξ -)| w * n H 1 • T -1 ζ ( ) | d .
The existence and the properties of w d , solution of (4.5), have been given in lemma 3.8. We adapt the proof of lemma 3.9. We first notice that | w(0)| ≤ c F 0 (0) by definition of c and by hypothesis on F 0 . Since w(0) = d +c F 0 (0), we have | w(0)| ≤ w(0). First case. We first assume | w(0)| < w(0). One considers, for L > 0, the open set U ρ,L of L-points. We remark that, once L 0 > 0 is chosen small enough, then for every

0 < L ≤ L 0 , for very d > 0, for every ζ ∈ U ρ,L , | w(ζ )| < w d (ξ ) with ξ = leng ρ (ζ ).
This is just a consequence of lemma 3.9. (For L > 0 small enough, leng ρ (ζ ) = |ζ |).

We now assume that there exist L 1 > 0 and

ζ 1 ∈ U ρ,L 1 such that | w(ζ 1 )| ≥ w d (ξ 1 )
,

ξ 1 = leng ρ (ζ 1 ). We recall that the mapping ζ ∈ R (1)
ρ → leng ρ (ζ ) is continuous and

we define χ = {L ∈ [L 0 , L 1 ] | there exists ζ ∈ U ρ,L , | w(ζ )| ≥ w d (leng ρ (ζ ))}
. This is a closed set bounded from below and we denote by L 2 ∈]L 0 , L 1 ] its infimum. This implies that:

• for every ζ ∈ U ρ,L 2 , | w(ζ )| ≤ w d (leng ρ (ζ )); • there exists ζ 2 ∈ U ρ,L 2 such that | w(ζ 2 )| = w d (ξ 2 ) and ξ 2 = leng ρ (ζ 2 ) = L 2 .
We take such a ζ 2 ∈ U ρ,L 2 . By (4.6),

| w(ζ 2 )| ≤ c F 0 (ξ 2 ) + ξ 2 0 [a + b(ξ 2 -)] | w H 1 • T -1 ζ 2 ( ) | d + c N ∑ n=1 ξ 2 0 F n (ξ 2 -)| w * n H 1 • T -1 ζ 2 ( ) | d
where H 1 is the shortest symmetrically contractile path associated with ζ 2 . We know by lemma 4.9 that leng ρ (H

1 • T -1 ζ 2 ( )) ≤ for every ∈ [0, ξ 2 ]
, while w d is real, positive and non-decreasing on R + . Therefore, We come back to the minor w of the formal series solution of the prepared equation (3.6). We already know that w can be analytically continued to the star-shaped domain • R (0) ⊂ C, with at most exponential growth of order 1 at infinity there (theorem 3.3). Said in other words, w can be interpreted as a holomorphic function on the principal sheet R (0) of the Riemann surface R. We claim that this function can be analytically continued to every R (0) -nearby sheets: this is the matter of the next theorem. 1. its minor w can be analytically continued to the Riemann surface R (1) . This provides a function in O(R (1) ) still denoted by w; 2. this function w has at most exponential growth of order 1 at infinity on R (1) . More precisely, for every ρ ∈]0, 1 5 ], there exist real positive constants A = A(ρ) > 0 and τ = τ(ρ) > 0 such that for every

| w(ζ 2 )| ≤ c F 0 (ξ 2 ) + ξ 2 0 [a + b(ξ 2 -)] w d ( ) d + c N ∑ n=1 ξ 2 0 F n (ξ 2 -) w * n d ( ) d .
ζ ∈ R (1) ρ , | w(ζ )| ≤ Ae τξ with ξ = leng ρ (ζ ); 3. moreover leng ρ (ζ ) ≤ 1 ρ | • ζ | + 1 ρ 1
ρ -2 and one can choose A = 4 and τ = 4

ρ 3
in the above estimates.

Proof. We begin this proof with a preliminary result which should be compared with lemma 3.2.

Lemma 4.12. There exists a real positive number M ρ,(1) > 0 such that for every polynomial q of degree ≤ 1, for every ζ ∈ R Proof. From lemma 3.2 and lemma 4.10, one sees that for every ζ ∈ R

ρ ,

leng ρ (ζ ) |P(- • ζ )| ≤ 1 ρ + 1 ρ 1 ρ -2 M ρ,(0)
. Then use the fact that ρ ∈]0, 1 5 ].

Holomorphy of w on R (1) Let r > 0 and ν > 0 be positive real numbers, U = U ρ,L ⊂ R (1) be the open set of L-points, L > 0, and

B r = { v ∈ O(U ), v ν ≤ r}.
The convolution equation (3.10) can be viewed as a fixed-point problem on B r and we set:

N : v ∈ B r → P -1 ( ∂ ) -1 * Q( ∂ ) v + f 0 + f 1 * v + f 2 * v * v .
By lemmas 4.12 and proposition 4.5,

N( v) ν ≤ M ρ,(1) -1 * Q( ∂ ) v + f 0 + f 1 * v + f 2 * v * v ν . By proposition 4.5, since Q( ∂ ) = -3 ∂ : 1 * Q( ∂ ) v ν ≤ 1 ν Q( ∂ ) v ν ≤ 1 Lν (ζ → Q(- • ζ )) 0 v ν ≤ 3L ν w ν .
The functions f 0 , f 1 , f 2 belong to M O(U ), therefore by proposition 4.5: lim

ν→∞ f i ν = 0, i = 0, 1, 2. Hence, N( v) ν ≤ r for ν > 0 large enough.
The same arguments shows that

N( v 1 ) -N( v 2 ) ν ≤ k v 1 -v 2 ν with k < 1, for v 1 , v 2 ∈ B r
and for ν > 0 large enough. The mapping N is thus contractive on the closed subset B r of the Banach space O(U ), . ν , for ν > 0 large enough. The contraction mapping theorem ensures the existence of a unique solution w ∈ B r for the fixed-point problem v = N( v). Since L and ρ can be arbitrarily chosen, we deduce (by uniqueness) that the minor w of the unique formal series w solution of (3.6) is a germ of holomorphic functions which can be analytically continued to R (1) .

Upper bounds

w = | f 0 |(z) + 3 z + | f 1 | w+| f 2 | w 2 , (4.7)
holomorphic at infinity and asymptotic to

| f 0 |(z) there. Remember that | f 0 |(z) = 392 625 1 z 2 , | f 1 |(z) = 4 z 2 , | f 2 |(z) = 1 2z 2 . Setting w(z) = H(t), t = z -1
, the above problem reads as a fixed-point problem,

H = N(H), N(H) = M ρ,(1) | f 0 |(t -1 ) + 3t + | f 1 |(t -1 ) H + | f 2 |(t -1 )H 2 . (4.8)
From homogeneity reasons, we introduce U = D(0, ρ 3 /4), we consider the Banach algebra O(U), where stands for the maximum norm, and we set B

Concluding remarks

The following comments rely on notions introduced in [10] to which the reader is referred.

It turns out from theorem 4.2 that the minor w can be analytically continued along any path of type γ θ ). Interpreting this result on R (1) where w is holomorphic with at most exponential growth of order 1 at infinity (theorem 4.2), we get: for every z ∈

1/2 0 = {ζ = ξ e iθ ∈ C | θ ∈] -π + θ 0 , θ 0 + 2π[, ξ ∈]0, 1/2[}.
• Π θ + τ(θ + ) ∩ • Π θ - τ(θ -) with |z| large enough, S θ + w(z) = S θ - w(z) + m ∑ k=1 γ k e -zζ w(ζ )dζ + C m+1 e -zζ w(ζ )dζ . (4.9)
One recognizes in this equation the very construction of the Stokes automorphism, detailed in [10]. The asymptotics at infinity of the integrals exponentially smaller vanishing behaviour. It will be shown in this course that the right-hand side of the equality (4.9) when letting m → ∞, is nothing but the Borel-Laplace sum of a "transseries" introduced and studied in chapters 5 and 6.

Some supplements

We end this chapter with some supplements to the that will be used later on. We denote by R (θ ,α) (L) the subset of the Riemann surface R defined as follows:

R (θ ,α) (L) = {Λ (1) ∈ R | λ = p • Λ belongs to R (θ ,α) (L) and Λ (0) = 0}.
We should note in passing that every path λ ∈ R (θ ,α) (L) can be lifted on R from 0 with respect to p. The following assertion is left as an exercise. 1≤ j≤m R (±) j ,θ with m = L . Also, for any m ∈ N , for any path γ of type γ θ ε with ε ∈ {+, -} j and 1 ≤ j ≤ m, the endpoint Γ (1) belongs to R (θ ,α) (m), where Γ is the lift of γ from 0 with respect to p on R.

In the above lemma, . is the ceiling function.

Remark 4.5. Notice that R (θ ,α) (L 1 ) ⊂ R (θ ,α) (L 2 ) when L 1 < L 2 . Also, since R (θ ,α) (L)
is open and connected in R, R (θ ,α) (L) inherits from R the structure of complex manifold, thus is a Riemann surface. Definition 4.24. We denote by R(θ,α) (L) ⊃ R the space of germs of holomorphic functions at the origin which can be analytically continued to the Riemann surface R (θ ,α) (L).

Example 4.1. The formal solution w of the prepared equation (3.6) has a minor w which belongs to R(θ,π/2) (L), for any direction θ ∈ {0, π} and any L ∈]0, 1]: this is a consequence of theorem 4.2.

Proposition 4.6. The space R(θ,α) (L) is a (non unitary) convolution algebra.

Proof. We just have to prove the stability by convolution product. It is shown in [10] that for any smooth path γ :

I = [0, 1] → C \ Z such that |γ(0)| < 1, one can find a symmetric Z-homotopy H : (s,t) ∈ I × I → H(s,t) = H t (s) whose initial path H 0 is H 0 : s ∈ [0, 1] → sγ(0) and whose endpoint path t ∈ [0, 1] → H t (1) coincides with γ.
Lifting every path H t from 0 on R with respect to p, one gets the mapping H : (s,t) ∈ I × I → H (s,t) = H t (s) ∈ R, which is continuous by the lifting theorem for homotopies, and the following diagram commutes: R H ↓ p

I × I -→ • R. H (4.10)
We recall from [10] how this symmetric Z-homotopy can be constructed. Pick a

C 1 function η : C → [0, 1] satisfying {ζ ∈ C | η(ζ ) = 0} = Z and consider the non-autonomous vector field X(ζ ,t) = η(ζ ) η(ζ ) + η γ(t) -ζ γ (t).
The path H t is obtained by deformation of the initial path H 0 through the flow of the vector field g :

(t 0 ,t, ζ ) ∈ [0, 1] 2 × C → g t 0 ,t (ζ ) ∈ C of X, precisely H t (s) = g 0,t H 0 (s) .
Let ζ be any point in R (θ ,α) (L). This point is the endpoint of a path Λ in R originating from 0 and whose projection λ = p • Λ belongs to R (θ ,α) (L). We forget the case where λ stays in D(0, 1) and, without loss of generality, we can assume that λ = λ 0 γ with λ 0 : s ∈ [0, 1] → sγ(0). Let us analyze the above symmetric Zhomotopy H constructed from γ and H 0 = λ 0 , and the associated mapping H . We would like to show that H (s,t) ∈ R (θ ,α) (L) for every (s,t) ∈ I × I. For this purpose we introduce the path H s : t ∈ I → H s (t) = H(s,t). We notice that H 0 ≡ 0 while for any s ∈]0, 1] :

1. H s (0) = H 0 (s), 2. dH s (t) dt = X H s (t),t , thus 0 < dH s (t) dt ≤ |γ (t)| and arg dH s (t) dt ∈] -α + θ , θ + α[.
Denoting by H 0 | [0,s] : s ∈ I → H 0 (s s) the restriction path, we see that the product of paths F s = H 0 | [0,s] H s has the following properties, for any s ∈]0, 1]:

1. F s is a path in • R originating from 0 and is piecewise C 1 ; 2. length(F s ) ≤ length(H 0 | [0,s] ) + length(H s ) ≤ length(λ ) ≤ L + 1, 3.
for every t ∈ [0, 1], the right and left derivatives (F s ) (t) do not vanish and arg(F s ) (t) ∈]α + θ , θ + α[.

Therefore F s belongs to R (θ ,α) (L) and as a consequence, H (s,t) belongs to R (θ ,α) (L) for every (s,t) ∈ I × I. We end the proof with the arguments used recalled in the proof of theorem 4.1.

Comments

As a rule in resurgence theory, one has to deal with endlessly continuable functions. This notion is defined in [2], a more general definition of which being given by Ecalle in [5,6]. The key point is the construction of endless Riemann surfaces [2,12]). For such an endless Riemann surface, one can define "nearby sheets" in the way we did in Sect. 4.2 and analogues of theorem 4.1 and proposition 4.6 can be stated.

Chapter 5

Transseries and formal integral for the first Painlevé equation

Abstract This chapter has two purposes. Our first goal is to construct the socalled "formal transseries solutions" for the prepared form associated with the first Painlevé equation, which will be used later on to get the truncated solutions : this is done in Sect. 5.3, after some preliminaries in Sect. 5.1 and Sect. 5.2. Our second goal is to build the formal integral for the first Painlevé equation and, equivalently, the canonical normal form equation to which the first Painlevé equation is formally conjugated. This is what we do in Sect. 5.4. These informations will be used in a next chapter to investigate the resurgent structure for the first Painlevé equation.

Introduction

We have seen in chapter 3 that the prepared equation (3.6) has a unique formal solution, from now on denoted by w (0,0) . This solution is 1-Gevrey and even Borel-Laplace summable in every directions apart from the directions kπ, k ∈ Z (theorem 3.

I j =] - 3 2 π, + 1 2 π[-jπ.
As said in Sect. 3.4.2.2, each w tri, j can be thought of as a section over ( I j of the sheaf A 1 of 1-Gevrey asymptotic functions, w tri, j ∈ Γ ( ( I j , A 1 ). These sections are asymptotic to the same 1-Gevrey series w (0,0) . Therefore the 1-coboundary w tri,1w tri,0 belongs to Γ (

( I 1 ∩ ( I 0 , A ≤-1 ), while w tri,2 -w tri,1 belongs to Γ ( ( I 2 ∩ ( I 1 , A ≤-1 )
, where A ≤-1 is the sheaf of 1-Gevrey flat germs. In other words, the 1-coboundaries

W 1,0 (z) = w tri,1 (z) -w tri,0 (z), - 3 2 π < arg(z) < - 1 2 π, |z| large enough, W 2,1 (z) = w tri,2 (z) -w tri,1 (z), - 5 2 π < arg(z) < - 3 2 π, |z| large enough, (5.1)
are exponentially flat functions of order 1 at infinity, and we deduce from equation (3.6) that W ( j+1), j , j = 0, 1, satisfies the linear ODE:

P(∂ )W ( j+1), j + 1 z Q(∂ )W ( j+1), j = f 1 (z)W ( j+1), j + f 2 (z)(w tri, j+1 + w tri, j )W ( j+1), j .
(5.2) 87 Question 5.1. Can we get more informations about W ( j+1), j ? In other words, are we able to analyze the Stokes phenomenon ?

We have already made some advances in reply to this query in Sect. 4.5.2, as an application of theorem 4.2. Here we return to the asymptotics. Denoting by T 1 the 1-Gevrey Taylor map, we set W ( j+1), j = T 1 ( ( I j+1 ∩ ( I j )W ( j+1), j . We have W ( j+1), j = 0 by construction but, more interestingly for our purpose and since T 1 is a morphism of differential algebras, we deduce from (5.2) that W ( j+1), j solves the problem P 0 W = 0, where P 0 stands for the second order linear differential operator deduced from the operator

P(∂ ) + 1 z Q(∂ ) -F(z, •
) by linearisation at w (0,0) :

P 0 = P(∂ ) + 1 z Q(∂ ) - ∂ F(z, w (0,0) ) ∂ w (5.3) = P(∂ ) + 1 z Q(∂ ) -f 1 (z) -2 w (0,0) (z) f 2 (z) = (∂ 2 -1) - 3 z ∂ + O(z -2 ).
For a moment, let us think of P 0 W = 0 as a linear ODE with holomorphic coefficients (thus we think of w (0,0) as a convergent series at infinity). The formal invariants for this this equation are governed by the Newton polygon at infinity N ∞ (P 0 ), drawn on Fig. 5.1.

The definition and properties of the Newton polygon are amply elaborated in [32], to which the reader is referred. We only mention that the valuation v ∞ defined there is the opposite of our valuation val defined by (3.1).

The polygon N ∞ (P 0 ) has a single non-vertical side of slope -1: this corresponds to the fact that W ( j+1), j , j = 0, 1, are exponentially flat functions of order 1 at infinity. The characteristic equation associated with this side is nothing but the equation

P(µ) = 0, P(µ) = µ 2 -1.
The polynomial P(µ) has two simple roots, µ 1 = -1 and µ 2 = 1. Therefore, from the theory of linear ODE [39,32], we expect for W 1,0 to behave like e µ 2 z z -τ 2 O(1) at infinity, and for W 2,1 to behave like e µ 1 z z -τ 1 O(1) at infinity. Pursuing in that direction, the coefficients τ 1 , τ 2 can be easily found : W = e µz z -τ w µ (z) solves the ODE (5.3) with P(µ) = 0 and

w λ ∈ C[[z -1 ]], only under the condition τ = Q(µ) P (µ) = - 3 2 .
Fig. 5.1 The Newton polygon at infinity N ∞ (P 0 ) associated with the linear operator (5.3).

1 0 2 3 -1
As a matter of fact, these behaviours are direct consequences of the analytic properties of the minor w (0,0) of w (0,0) . In particular, λ 1 = -µ 1 and λ 2 = -µ 2 are precisely the so-called seen singularities of w (0,0) , cf. theorem 4.2.

The differential equation P 0 W = 0 has thus its general formal solution under the form W = U 1 e µ 1 z z -τ 1 w µ 1 + U 2 e µ 2 z z -τ 2 w µ 2 and, as we will see later on, both w µ 1 and w µ 2 are 1-Gevery series whose minors have the same properties than w (0,0) .

However, the expectation that W 1,0 could be obtained from U 1 e µ 1 z z -τ 1 w µ 1 by Borel-Laplace summation for some well-chosen U 1 ∈ C is wrong. Indeed, this would mean that w tri,1 = S I 1 w (0,0) +U 1 e µ 1 z z -τ 1 w µ 1 , thus w (0,0) +U 1 e µ 1 z z -τ 1 w µ 1 is a formal solution of (3.6). This is not the case because of the nonlinearity of (3.6) and to the very nature of the Riemann surface R (1) on which w (0,0) can be analytically continued (theorem 4.2). This raises the question: Question 5.2. can we define an analogue of the general formal solution for the nonlinear equation (3.6) ?

The answer is given by the notion of "formal integral" which we now introduce.

Formal integral : setting

Notations

It will be useful in the sequel to fix customary notations.

Definition 5.1. We suppose n ∈ N , k, h ∈ N n , a, b ∈ C n . • If k = (k 1 , • • • , k n ), then |k| = k 1 + • • • + k n . • If a = (a 1 , • • • , a n ) or a = t (a 1 , • • • , a n ), then a k = a k 1 1 • • • a k n n . • If b = (b 1 , • • • , b n ), then a.b = a 1 b 1 + • • • + a n b n .
• We denote by e j the j-th unit vector of C n .

Setting

Single level 1 ODE

To introduce the reader to the notion of Ecalle's formal integral [19], it will be useful to skip a moment from the ODE (3.6) to a more general one 1 with the same kind of properties. Namely we introduce

P(∂ )w + 1 z Q(∂ )w = F(z, w) (5.4)
P(∂ ) = n ∑ m=0 α n-m ∂ m ∈ C[∂ ] , Q(∂ ) = n-1 ∑ m=0 β n-m ∂ m ∈ C[∂ ]
1 Though far from the more general. For instance in (5.4) one could replace F(z, w) by

F(z, w, ∂ w, • • • , ∂ n-1 w), with F holomorphic in a neighbourhood of (∞, 0) in C × C n-1
, see exercise 3.1. We refrain of doing that only for a matter of simplicity. See [19] for more general results.

with n ∈ N . We assume that P is a polynomial of degree n, that is α 0 = 0, and that F(z, w) is holomorphic in a neighbourhood of (∞, 0) in C 2 with the condition

∂ m F ∂ w m (z, 0) = O(z -2
), m ∈ N. (See exercise 3.1). We will add other assumptions to guarantee that the ODE (5.4) has a single level 1 at infinity. When assuming furthermore that α n = 0, what have been said in Sect. 5.1 can be applied as well for (5.4). The equation (5.4) has a unique formal solution w 0 ∈ C[[z -1 ]] and val w 0 ≥ 2. The Newton polygon at infinity N ∞ (P 0 ) associated with the linear differential operator

P 0 = P(∂ ) + 1 z Q(∂ ) - ∂ F ∂ w (z, w 0 ) deduced from the operator P(∂ ) + 1 z Q(∂ ) -F(z, •
) by linearisation at w 0 , has still a single nonvertical side of slope -1 and the characteristic equation associated with this single side remains the equation P(µ) = 0. Since α n = 0, the roots of the characteristic equation do not vanish. We will also assume that the polynomial

µ → P(µ) = n ∑ m=0 α n-m µ m = α 0 (µ -µ 1 ) • • • (µ -µ n ) has only simple roots µ = µ i , i = 1, • • • , n.
The following definitions are adapted from [3,19]. Definition 5.2. Let {µ i } be the set of the roots of the polynomial P(µ), and we set

λ i = -µ i , i = 1, • • • , n. The complex numbers λ 1 , • • • , λ n are
called the multipliers of the ODE (5.4). The ODE (5.4) is said to have a single level 1 at infinity when the multipliers are all nonzero. One says that the multipliers are non-resonant when they are rationally independent, that is linearly independent over Z. The multipliers are positively resonant when there exists

k reson = (k 1 , • • • , k n ) ∈ N n \ {0} so that λ λ λ .k reson = 0, where λ λ λ = (λ 1 , • • • , λ n ) ∈ (C ) n
The number |k reson | + 1 is the order of the resonance, since the positive resonance brings semi-positively resonances, that is relationships of the type λ λ λ .(k reson + e j ) = λ j for any j ∈ [1, n].

We mention that the following constants are properly defined, since P has only simple roots:

τ i = Q(-λ i ) P (-λ i ) , i = 1, • • • , n. (5.5)
From the theory of linear differential equations (see [34,32], see also [4,39]), we notice that the linear equation P(∂ )w + 1 z Q(∂ )w = 0 has a formal general solution under the form

w(z) = n ∑ i=1 v i (z)y i (z).
(5.6)

In (5.6), y i (z) = U i e -λ i z z -τ i , U i ∈ C, stands for the general solution of the differential

equation y i + λ i + τ i z y i = 0, while v i ∈ C[[z -1 ]
] is invertible and is determined uniquely up to normalization.

Companion system, prepared form

Formal integrals have more natural foundations when differential equations of order one are considered. We therefore translate the ODE (5.4) into a one order ODE of dimension n by introducing w =

     w 1 w 2 . . . w n      =      w w . . . w (n-1)     
. We get the companion system

∂ w + Aw = f(z, w), (5.7
)

with A =       0 -1 • • • 0 . . . . . . . . . . . . . . . 0 -1 α n α 0 + β n zα 0 • • • • • • α 1 α 0 + β 1 zα 0       and f(z, w) =      0 . . . 0 F(z, w 1 )/α 0      . Since (5.4) has a unique formal solution w 0 ∈ C[[z -1 ]
], val w 0 ≥ 2, we may remark that (5.7) has a unique formal solution w 0 ∈ C n [[z -1 ]] as well, and in fact

w 0 ∈ z -2 (C[[z -1 ]]) n . Lemma 5.1. There exists T 0 (z) ∈ GL n (C{z -1 }[z]
) so that the meromorphic gauge transform w = T 0 (z)v brings (5.7) into the prepared form

∂ v + B 0 v = g(z, v), B 0 =    λ 1 + τ 1 z • • • 0 . . . . . . . . . 0 • • • λ n + τ n z    , (5.8)
with g a C n -valued function, holomorphic in a neighbourhood of (∞, 0) and

g(z, v) = O(z -2 ) + O( v 2 ) when z → ∞ and v → 0.
The prepared form (5.8) has a unique formal solution

v 0 ∈ (C[[z -1 ]]) n and v 0 ∈ z -2 (C[[z -1 ]]) n .
Proof. The proof is based on classical ideas for linear ODEs (see [34,32], see also [4,39,15]). Looking at (5.6), we compare (5.7) with the linear equation

∂ u + B 0 u = 0, B 0 =    λ 1 + τ 1 z • • • 0 . . . . . . . . . 0 • • • λ n + τ n z    = Λ + 1 z L, (5.9)
whose general solution (holomorphic on C) is given in term of the fundamental matrix solution z -L e -zΛ ,

u(z) = z -L e -zΛ U = ⊕ n i=1 z -τ i e -zλ i U, U ∈ C n .
(5.10)

We remark that

e -λ z z -τ (m) = e -λ z z -τ m ∑ j=0 m j (-λ ) m-j (-τ) j z j (5.11)
for (λ , τ) ∈ C 2 and m ∈ N, where (-τ) j = j! -τ j mimics the Pochhammer symbol:

(-τ) 0 = 1 and (-τ) j = (-1) j τ(τ + 1) • • • τ + j -1 for j ≥ 1.

(5.12)

We thus set the meromorphic gauge transform w = T 0 (z)v with T 0 (z) ∈ GL n (C{z -1 }[z]) of the form:

T 0 (z) =        1 • • • 1 -λ 1 -τ 1 z • • • -λ n -τ n z . . . . . . ∑ n-1 j=0 n -1 j (-λ 1 ) n-1-j (-τ 1 ) j z j • • • ∑ n-1 j=0 n -1 j (-λ n ) n-1-j (-τ n ) j z j       
.

(5.13) By its very definition, this gauge transform brings (5.7) into the differential equation:

∂ v = -T -1 0 (∂ T 0 ) + T -1 0 AT 0 v + T -1 0 f(z, T 0 v) (5.14) = -B 0 v + g(z, v)
where g has the properties described in the lemma. The fact that (5.8) has a unique formal solution

v 0 ∈ (C[[z -1 ]]) n is obvious.
Example 5.1. We have already seen that the companion system associated with (3.6) is (3.9). The gauge transform

w = T 0 (z)v, T 0 (z) = 1 1 -1 + 3 2z 1 + 3 2z , brings (3.9)
into the prepared form:

∂ v + 1 -3 2z 0 0 -1 -3 2z v = 15 8z 2 -1 -1 1 1 v + 1 2 -F(z, v 1 + v 2 ) F(z, v 1 + v 2 ) . (5
A 0 =       0 -1 • • • 0 . . . . . . . . . . . . . . . 0 -1 p n (z) • • • • • • p 1 (z)       with p n , • • • , p 1 ∈ C{z -1 }, p n (z) = α n α 0 + β n zα 0 + O(z -2 ), • • • , p 1 (z) = α 1 α 0 + β 1 zα 0 + O(z -2
). The system ∂ y + A 0 y = 0 is the companion system for the one-dimensional homogeneous ODE of order n,

∂ n y + p 1 (z)∂ n-1 y + • • • + p n (z)y = 0, (5.16
)

whose general solution is y(z) = ∑ n i=1 U i e -λ i z z -τ i , (U 1 , • • • ,U n ) ∈ C n .

Normal form, formal reduction

We have previously reduced the companion system (5.7) to a prepared form through a meromorphic gauge transform. Under some conditions, one can go further, but through formal transformations, in the spirit of the Poincaré-Dulac theorem [3] and the classification up to formal conjugation.

Proposition 5.1. We consider the ODE (5.8) and we assume that the multipliers λ 1 , • • • , λ n are non-resonant. Then there exists a formal transformation v = T (z, u),

T (z, u) = ∑ k∈N n u k v k (z), v k (z) ∈ (C[[z -1 ]]) n , (5.17)
which formally transforms (5.8) into the linear normal form equation ∂ u + B 0 u = 0. In (5.17), v 0 stands for the unique formal solution of (5.8); for j = 1, . . . , n, v e j is uniquely determined when one prescribes its constant term to be equal to e j ; then the formal series v k are unique for |k| > 1.

We will see in the sequel how this proposition can be shown. Here, we rather concentrate on its consequences.

One can refer to, e.g., [31,5] for a proof that extend to possibly nilpotent cases (but with no resonances), and to [19] for a very general frame.

We know that the general solution of the normal form

∂ u + B 0 u = 0 is u(z) = ⊕ n i=1 z -τ i e -zλ i ( t U), U = (U 1 , • • • ,U n ) ∈ C n .
Through the action of the formal transformation v = T (z, u), this provides the following general formal solution for the ODE (5.8):

v(z, U) = ∑ k=(k 1 ,••• ,k n )∈N n n ∏ i=1 (U i z -τ i e -zλ i ) k i v k (z) = ∑ k∈N n U k e -λ λ λ .kz z -τ τ τ.k v k (z) (5.18) with λ λ λ = (λ 1 , • • • , λ n ) ∈ (C ) n and τ τ τ = (τ 1 , • • • , τ n ) ∈ C n .
Definition 5.3. The formal series (5.18) is called the formal integral of (5.8).

Of course, one can obtain the formal integral w(z, U) of (5.7) as well, by the gauge transform w = T 0 (z) v, with T 0 (z) given by (5.13). When finally returning to the n-th order ODE (5.4) of dimension 1 we started with, one gets its formal integral. Definition 5.4. We assume that the multipliers are non-resonant. The formal integral w(z, U) of the ODE (5.4) is defined by:

w(z, U) = ∑ k∈N n U k e -λ λ λ .kz z -τ τ τ.k w k (z), w k (z) = v k (z).(1, • • • , 1) ∈ C[[z -1 ]], = Φ(z,U 1 e -λ 1 z z -τ 1 , • • • ,U n e -λ n z z -τ n ) (5.19) with Φ(z, u) = ∑ k∈N n u k w k (z) ∈ C[[z -1 , u]].
The formal transformation w = Φ(z, u) formally transforms (5.4) into the normal form equation ∂ u + B 0 u = 0.

The formal integral (5.19), thus depending on the maximal n free parame-

ters U = (U 1 , • • • ,U n ) ∈ C n ,
plays the role of the general formal solution for the ODE (5.4) of order n. Formal integrals can be defined as well for difference and differential-difference equations, see, e.g. [19,31]. This notion has been enlarged for nonlinear partial differential equations in [35].

Remark 5.2. Although working at the formal level, one may wonder what is the chosen branch when we write z -τ τ τ.k . As a matter of fact, this is not relevant at this stage since moving from a determination to another one just translates into rescaling the free parameter U.

Remark 5.3. Introducing V k = U k e -λ λ λ .kz z -τ τ τ.k , we remark the identity:

∂ z V k w k = ∂ z - n ∑ i=1 (λ i + τ i z )u i ∂ u i u k w k | u=V .
Looking at the equality

w(z, U) = Φ(z,U 1 e -λ 1 z z -τ 1 , • • • ,U n e -λ n z z -τ n ) (5.20)
and since the formal integral (5.19) solves the differential equation (5.4), one deduces that Φ satisfies:

P ∂ z - n ∑ i=1 (λ i + τ i z )u i ∂ u i Φ + 1 z Q ∂ z - n ∑ i=1 (λ i + τ i z )u i ∂ u i Φ = F(z, Φ). (5.21)

Formal integral, general considerations

Under convenient hypotheses, we have previously introduced the formal integral for the ODE (5.4), that is a n-parameters formal expansion of the form

w(z, U) = ∑ k∈N n U k e -λ λ λ .kz z -τ τ τ.k w k (z), λ λ λ , τ τ τ ∈ C n , (5.22)
Let us start with (5.22) and investigate the conditions to impose on the w k 's in order for (5.22) to be formally solution of (5.4).

We could start with (5.21) as well.

Using the identity (5.11) for m ∈ N, one obtains from (5.22):

w (m) = ∑ |k|≥0 U k m ∑ p=0 m p (e -λ λ λ .kz z -τ τ τ.k ) (p) w (m-p) k = ∑ |k|≥0 U k e -λ λ λ .kz z -τ τ τ.k T k,m+1 (w k )
where T 0,m+1 (w 0 ) = w (m) 0 and, more generally for k ∈ N 2 ,

T k,m+1 (w k ) = m ∑ p=0 m p p ∑ j=0 p j (-λ λ λ .k) p-j (-τ τ τ.k) j z j w (m-p) k = m ∑ j=0 m j (-τ τ τ.k) j z j m-j ∑ q=0 m -j q (-λ λ λ .k) m-j-q w (q) k ,
that is also

T k,m+1 (w k ) = m ∑ j=0 m j (-τ τ τ.k) j z j (-λ λ λ .k + ∂ ) m-j w k . (5.23)
In what follows we will simply write T k,m+1 instead of T k,m+1 (w k ). We introduce the notation V k = U k e -λ λ λ .kz z -τ τ τ.k and we notice that for every

k 1 , k 2 ∈ N n , V k 1 V k 2 = V k 1 +k 1 .
On the one hand,

P(∂ )w = ∞ ∑ k=0 V k n ∑ m=0 α n-m T k,m+1 = ∑ |k|≥0 V k p k (∂ )w k (5.24)
where for |k| ≥ 0,

p k (∂ ) = n ∑ m=0 α n-m (-λ λ λ .k + ∂ ) m + n ∑ m=1 α n-m m ∑ j=1 m j (-τ τ τ.k) j z j (-λ λ λ .k + ∂ ) m-j .
In other words, for |k| ≥ 0,

p k (∂ ) = P(-λ λ λ .k + ∂ ) + n ∑ j=1 1 z j -τ τ τ.k j P ( j) (-λ λ λ .k + ∂ ). (5.25)
Similarly

Q(∂ )w = ∑ |k|≥0 V k q k (∂ )w k (5.26) with q k (∂ ) = Q(-λ λ λ .k + ∂ ) + n-1 ∑ j=1 1 z j -τ τ τ.k j Q ( j) (-λ λ λ .k + ∂ ).
(5.27)

On the other hand we consider the Taylor expansion of F(z, w(z, U)) at w 0 , namely

F(z, w) = F(z, w 0 ) + ∑ ≥1 ∑ |k|≥1 V k w k ! ∂ F(z, w 0 ) ∂ w . (5.28)
We observe that for every ∈ N ,

∑ |k|≥1 V k w k = ∑ |p|≥ V p ∑ p 1 +•••+p =p |p i |≥1, 1≤i≤ w p 1 • • • w p . (5.29)
As a result, equation (5.28) reads

F(z, w) = F(z, w 0 ) + ∑ ≥1 |p|≥ V p ∑ p 1 +•••+p =p |p i |≥1, 1≤i≤ w p 1 • • • w p ! ∂ F(z, w 0 ) ∂ w . (5.30)
Finally, plugging the formal expansion (5.22) into the differential equation (5.4), using the identities (5.24), (5.26), (5.30) and identifying the powers V k , one gets the next lemma 5.2 which justifies the following definition.

Definition 5.5. For k ∈ N n , we define

P k (∂ ) = P(-λ λ λ .k + ∂ ), (5.31) Q k (∂ ) = -τ τ τ.kP (-λ λ λ .k + ∂ ) + Q(-λ λ λ .k + ∂ ) R k (∂ ) = n-2 ∑ j=0 1 z j -τ τ τ.k j + 2 P ( j+2) (-λ λ λ .k + ∂ ) + -τ τ τ.k j + 1 Q ( j+1) (-λ λ λ .k + ∂ ) .
(5.32)

For k ∈ N n , we denote by D k = D k (w 0 ) the linear differential operator

D k = P k (∂ ) + 1 z Q k (∂ ) + 1 z 2 R k - ∂ F(z, w 0 ) ∂ w
where w 0 satisfies P(∂

)w 0 + 1 z Q(∂ )w 0 = F(z, w 0 ).
For k ∈ N n , we denote by P k = P k (w 0 ) the linear differential operator U k e -λ λ λ .kz z -τ τ τ.k w k (z) (5.34) solves (5.4) if and only if :

P k = P(-λ λ λ .k + ∂ ) + 1 z Q(-λ λ λ .k + ∂ ) - ∂ F(z, w 0 ) ∂ w . ( 5
P(∂ )w 0 + 1 z Q(∂ )w 0 = F(z, w 0 ), (5.35
)

D e i w e i = 0 (5.36)
with e i the i-th vector of the canonical base of C n , and for |k| ≥ 2,

D k w k = ∑ k 1 +•••+k =k |k i |≥1, ≥2 w k 1 • • • w k ! ∂ F(z, w 0 ) ∂ w . (5.37)
Remark 5.4. Notice that in lemma 5.2 we have neither supposed that

λ λ λ = (λ 1 , • • • , λ n ) are the multipliers, nor that τ τ τ = (τ 1 , • • • , τ n ) are such that τ i = Q(-λ i ) P (-λ i ) , i = 1, • • • , n.
However, these conditions will come in the next section.

Example 5.2. We consider equation (3.6) where n = 2, P(∂

) = ∂ 2 -1, Q(∂ ) = -3∂ .
Then, for every k ∈ N 2 ,

P k (∂ ) = ∂ 2 -2λ λ λ .k∂ + (λ λ λ .k) 2 -1, (5.38) Q k (∂ ) = (3 + 2τ τ τ.k)(-∂ + λ λ λ .k), R k (∂ ) = τ τ τ.k(τ τ τ.k + 4).
In particular, taking λ λ λ = (1, -1) (the zeros of

ζ → P(-ζ )) and τ τ τ = - 3 2 , - 3 2
(we take the values given by (5.5)), then writing k = (k 1 , k 2 ):

P k (∂ ) = ∂ 2 -2(k 1 -k 2 )∂ + (k 1 -k 2 ) 2 -1, (5.39) Q k (∂ ) = 3(1 -k 1 -k 2 )(-∂ + k 1 -k 2 ), R k (∂ ) = 9 4 (k 1 + k 2 ) k 1 + k 2 - 8 3 .
We eventually mention some identities for later purposes, the proof of which being left as an exercise.

Lemma 5.3. The operators P k and D k given by definition 5.5 satisfy the identities: for any k, k 1 , k 2 ∈ N n , e -λ λ λ .k 1 z P k 1 e λ λ λ .k 1 z = e -λ λ λ .k 2 z P k 2 e λ λ λ .k 2 z , z -τ τ τ.k D k = P k z -τ τ τ.k and (e -λ λ λ .k 1 z z -τ τ τ.k 1 )D k 1 (e -λ λ λ .k 1 z z -τ τ τ.k 1 ) -1 = (e -λ λ λ .k 2 z z -τ τ τ.k 2 )D k 2 (e -λ λ λ .k 2 z z -τ τ τ.k 2 ) -1 .

Setting W k = z -τ τ τ.k w k for k ∈ N n and the w k given by lemma 5.2, one has P e i W e i = 0, i = 1, 2 while and for |k| ≥ 2,

P k W k = ∑ k 1 +•••+k =k |k i |≥1, ≥2 W k 1 • • •W k ! ∂ F(z, w 0 ) ∂ w .
(5.40)

First Painlevé equation and transseries solutions

We partly describe in this section the contains of lemma 5.2 for the prepared form equation (3.6) associated with the first Painlevé equation. Thus

n = 2, P(∂ ) = ∂ 2 -1, Q(∂ ) = -3∂ and F(z, w) = f 0 (z) + f 1 (z)w + f 2 (z)w 2 .
Also, we will for the moment specialise our study to only one-parameter formal expansions, that is we will assume that either U 1 = 0 or U 2 = 0 in (5.34). This study will be enough to get the truncated solutions. We will keep on our study of the formal integral associated with (3.6) in Sect. 5.4 where will we see the effects of resonances.

Transseries solution -statement

This section will be devoted to proving the following proposition.

Proposition 5.2. We consider the prepared ODE (3.6). We set λ λ λ = (λ 1 , λ 2 ) = (1, -1) where the λ i 's are the multipliers, that is the roots of the polynomial ζ → P(-ζ ).

We set

τ τ τ = (τ 1 , τ 2 ) = - 3 2 , - 3 2
, where

τ i = Q(-λ i ) P (-λ i ) , i = 1, 2.
Then for each i = 1, 2, there exists a formal one-parameter solution of (3.6) in the graded algebra k∈N z -τ i k e -λ i kz C[[z -1 ]] of the form:

w(z,Ue i ) = ∞ ∑ k=0 U k e -λ i kz z -τ i k w ke i (z), w ke i ∈ C[[z -1 ]]. (5.41)
We have val w ke i = 2(k -1) and the formal series (5.41) is unique once one fixes the normalization of w e i to be w e i (z

) = 1 + O(z -1 ). Then w ke i ∈ R[[z -1 ]] and w ke i (z) = k 12 k-1 z -2(k-1) (1 + O(z -1
)) for every k ≥ 1. Furthermore changing the normalization of w e i is equivalent in rescaling the parameter U ∈ C. Eventually, w ke 1 (z) = w ke 2 (-z) for every k ≥ 0.

Definition 5.6. The series (5.41) is called a formal transseries. The terms e -λ i kz z -τ i k are (log-free) transmonomials. The formal series w ke i are called the ke i -th series of the transseries. We set W ke i = z -τ i k w ke i .

Remark 5.5. The term "transseries" is due to Ecalle [20]. These are objects that are widely used in resurgence theory, see, e.g. [9,36,28,29]. More details on transseries can be founded in [19,20,7,8]. Transseries are also common objects in theoretical physics : these are the so-called "multi-instanton expansions", see e.g. [40,25,26,27,33,24,1,16,17,18].

In quantum mechanics or quantum field theory, an instanton action (the terminology of which is due to Gerard 't Hooft) is a classical solution of the equations of motion, with a finite and non-zero action. A well-known instanton effect in quantum mechanics is given by a particle in a double well potential. The tunneling effect provides a non-zero probability that the particle crosses the potential barrier. This gives rise to a tunneling amplitude proportionnal to the instanton e -S/h where S is the instanton action, h being the Planck constant or the coupling constant. For the bound states, this translates into the fact that they can be described at a formal level by a multi-instanton expansion, that is a transseries of the form ∑ k≥0 E k (h)e -kS/h where the perturbative fluctuations E k (h) are formal expansions with respect to h. The bound states are deduced from the multi-instanton expansion by (median) Laplace-Borel summation, see [38,10,11,12,13,14,22,23].

For later use, we mention a lemma that result from proposition 5.2 and lemma 5.3.

Lemma 5.4. Under the conditions of proposition 5.2 and for any k ∈ N 2 , the (socalled) general formal solution of the linear differential equation

P k ( w 0 ) W = 0 is W = e λ λ λ .kz C 1 e -λ 1 z W e 1 + C 2 e -λ 2 z W e 2 , C 1 ,C 2 ∈ C. For any k ∈ N 2 the (so- called) general formal solution of the linear differential equation D k ( w 0 ) w = 0 is w(z) = e λ λ λ .kz z τ τ τ.k C 1 e -λ 1 z W e 1 +C 2 e -λ 2 z W e 2 , C 1 ,C 2 ∈ C.

Transseries solution -proof

A useful lemma

We start with the following lemma which will be useful in the sequel. Lemma 5.5. We suppose n, N ∈ N . We consider the ordinary differential equation

P(∂ )w + 1 z R(∂ )w = f (z), f (z) = f N z -N (1 + O(z -1 )) ∈ z -N C[[z -1 ]], f N = 0 with P(∂ ) = ∑ n m=0 α n-m ∂ m ∈ C[∂ ], α n = 0, R(∂ ) = ∑ n-1 m=0 γ n-m (z)∂ m ∈ C[[z -1 ]][∂ ]. This ODE has a unique solution w in C[[z -1 ]], moreover val w = val f and w(z) = f N P(0) z -N (1 + O(z -1 )).
Proof. In the valuation ring C[[z -1 ]] we consider the following map :

N : C[[z -1 ]] → C[[z -1 ]] w → 1 P(0) f (z) -P(∂ ) -P(0) w - 1 z R(∂ )w .
(Remember that P(0) = α n is nonzero). From the hypotheses made one easily observes that

N(C[[z -1 ]]) ⊂ z -1 C[[z -1 ]] while, for every p ∈ N , if u, v ∈ z -p C[[z -1 ]], then N(u) -N(v) ∈ z -p-1 C[[z -1 ]]. This means that N is contractive in C[[z -1 ]], thus the fixed point problem w = N(w) has a unique solution w = lim p→∞ N p (0) in C[[z -1 ]]. Since N(0) = f (z)/P(0) one gets w(z) = f N P(0) z -N (1 + O(z -1
)).

Proof of proposition 5.2

We precise as an introduction that the assertion

w ke i ∈ R[[z -1 ]
] is just a consequence of the realness of equation (3.6). The relationships w (0,k) (z) = w (k,0) (-z) for every k ≥ 0, come from the property of equation (3.6) to be invariant under the change of variable z → -z and to the chosen normalization of w e i , i = 1, 2.

The return of the formal solution

We remark that w 0 = w (0,0) has to solve (5.35) which is nothing but the equation (3.6) one started with. In particular we know that this equation has a unique formal solution w 0 ∈ C[[z -1 ]] which has been investigated in the previous chapters.

In what follows, one will always replace w 0 by this formal solution w 0 . We mention the following obvious fact, essentially due to the property that val w 0 ≥ 2 and that for every = 0, 1, 2, ∂ F(z,0)

∂ w ∈ z -2 C{z -1 }. (
(z) = ∑ l≥2 a 0,l z -l ∈ C[[z -1 ]] is the formal solution of (3.6), then for every = 0, 1, 2, ∂ F(z, w 0 ) ∂ w ∈ C[[z -1 ]
] has valuation 2, and vanishes identically for every ≥ 3. Also, ∂ F(z, w 0 ) ∂ w = -4z -2 + z -2 w 0 is even and its coefficients are all real negative, and where

∂ 2 F(z, w 0 ) ∂ w 2 = z -2 .

The cases |ke

D e 1 = P e 1 (∂ ) + 1 z Q e 1 (∂ ) + 1 z 2 R e 1 - ∂ F(z, w 0 ) ∂ w with P e 1 (∂ ) = P(-λ 1 + ∂ ) = P(-λ 1 ) + P (-λ 1 )∂ + P (-λ 1 ) 2! ∂ 2 Q e 1 (∂ ) = -τ 1 P (-λ 1 + ∂ ) + Q(-λ 1 + ∂ ) R e 1 = τ 1 (τ 1 + 4) Assuming that w e 1 ∈ C[[z -1 ]],
one observes that the right-hand side of (5.42) has valuation less or equal to (val w e 1 ) -2, because of lemma 5.6. In order to get a non identically vanishing solution, one thus has to impose the condition P(-λ 1 ) = 0. Following our conventions, we take λ 1 = 1. The same reasoning leads to impose furthermore that -τ 1 P (-λ 1 ) + Q(-λ 1 ) = 0, thus τ 1 = -3 2 . Therefore, where where w e 1 and w e 2 are uniquely determined by their given normalization

P e 1 (∂ ) = ∂ 2 -2∂ , Q e 1 (∂ ) = 0, R e 1 (∂ ) = -
D e 2 = P e 2 (∂ ) + 1 z Q e 2 (∂ ) + 1 z 2 R e 2 -∂ F(z, w 0 ) ∂ w whereas P e 2 (∂ ) = ∂ 2 + 2∂ , Q e 2 (∂ ) = 0, R e 2 (∂ ) = -15 4 .
w e i = 1 + O(z -1 ). Moreover w e i ∈ R[[z -1 ]
] and w e 2 (z) = w e 1 (-z). Furthemore, if w 0 (z) = ∑ l≥0 a 0,l z -l and w e 1 (z) = ∑ l≥0 a e 1 ,l z -l , the following quadratic recursion relation is valid:

     a e 1 ,0 = 1, a e 1 ,l = 1 8l -(2l -1) 2 a e 1 ,l-1 + 4 l-1 ∑ p=0 a e 1 ,p a 0,l-p-1 , l = 1, 2, • • • (5.44)
Proof. We only examine (5.42). We look at this equation in the space of normalized formal series

C[[z -1 ]], namely    (∂ -2)∂ w e 1 = 15 4 1 z 2 + ∂ F(z, w 0 ) ∂ w w e 1 w e 1 ∈ C[[z -1 ]], w e 1 = 1 + O(z -1
).

(5.45)

We remark that the restriction of the derivation operator ∂ to the maximal ideal

z -1 C[[z -1 ]] is a bijective operator between z -1 C[[z -1 ]] and z -2 C[[z -1 ]]; we denote by ∂ -1 the inverse operator, ∂ → z -1 C[[z -1 ]] z -2 C[[z -1 ]]. ← ∂ -1
We transform (5.45) into the equation -2∂ w e 1 = -∂ 2 + 15 This means that the map

N : C[[z -1 ]] → C[[z -1 ]] w e 1 → 1 - 1 2 ∂ -1 -∂ 2 + 15 4 1 z 2 + ∂ F(z, w 0 ) ∂ w w e 1
is well defined and the problem (5.45) is equivalent to the fixed-point problem w e 1 = N(w e 1 ). One easily checks that the map N is contractive in C[[z -1 ]] so that the fixed point problem w e 1 = N(w e 1 ) has a unique solution

w e 1 in C[[z -1 ]].
From the fact that (5.42) is a homogeneous equation, one immediately concludes that U 1 w e 1 , U 1 ∈ C, provides a one-parameter family of formal solutions. The proof for the quadratic recursion relation (5.44) is left to the reader (see also [24,1]).

Remark 5.6. 1. The Newton polygon at infinity N ∞ (D e 1 ) drawn on Fig. 5.2, has one horizontal side that corresponds to the operator -2∂ . General nonsense in asymptotic theory (see [32], or [6,30]) provides the existence of the formal (normalized) series solution w e 1 . The other (normalized) formal solution associated with the side of slope -1 is e 2z w e 2 (see lemma 5.4) which, in our frame, is already incorporated in the other transseries solution. 2. From lemma 5.6 or (5.44), one easily shows that

w e 1 (z) = 1 - 1 8 z -1 + 9 128 z -2 - 341329 1920000 z -3 + • • •
is a real formal expansion, with coefficients that alternate in sign.

The cases |ke i | ≥ 2

Lemma 5.8. For any k = ke i , i = 1, 2 and k ≥ 2, equation (5.37) has a unique formal solution

w ke i = w ke i in C[[z -1 ]]. Moreover val w ke i = 2(k -1).
Furthermore, when considering U w e i instead of w e i for the solution of (5.36), then the unique solution of (5.37

) at rank k = ke i , k ≥ 2, is U k w ke i . Also, w ke i ∈ R[[z -1 ]],
w ke i (z) = k 12 k-1 z -2(k-1) (1 + O(z -1 )) and w (0,k) (z) = w (k,0) (-z) for every k ≥ 2.
Eventually, writing w ke 1 (z) = ∑ l≥0 a ke 1 ,l z -l , the coefficients are governed the following quadratic recursion relations, for every k ≥ 2:

                 a ke 1 ,0 = a ke 1 ,1 = 0, (k 2 -1)a ke 1 ,l = k(3k -2l -1)a ke 1 ,l-1 -1 4 (3k -2l) 2 a ke 1 ,l-2 + ∑ l-2 p=0   ake 1 ,p a 0,l-p-2 + 1 2 ∑ k 1 +k 2 =k k 1 ≥1, k 2 ≥1 a k 1 e 1 ,p a k 2 e 1 ,l-p-2    , l = 2, 3, • • • (5.46)
Proof. We only examine the case k = ke 1 , k ≥ 2. The proof is done by induction on k. We first consider equation (5.37) for k = 2:

D 2e 1 w 2e 1 = w 2 e 1 2! ∂ 2 F(z, w 0 ) ∂ w 2 , (5.47) with D 2e 1 = P 2e 1 (∂ ) + 1 z Q 2e 1 (∂ ) + 1 z 2 R 2e 1 -∂ F(z, w 0 ) ∂ w .
We know that P 2e 1 (0) = 3 is nonzero since, by (5.31), P 2e 1 (∂ ) = P(-2λ 1 + ∂ ) = ∂ 2 -4∂ + 3. Using lemma 5.6, one sees that lemma 5.5 can be applied to (5.47) and this provides a unique solution

w 2e 1 ∈ C[[z -1 ]].
Its valuation is 2 and explicit calculation gives:

w 2e 1 (z) = 1 6 z -2 - 11 72 z -3 + 53 192 z -4 + • • • , w 2e 2 (z) = w 2e 1 (-z).
One easily checks that replacing w e 1 by U w e 1 implies changing w 2e 1 into U 2 w 2e 1 .

We now assume that the properties of lemma 5.8 are true for every 2 ≤ k ≤ K -1. When considering equation (5.37) for K one gets :

D Ke 1 w Ke 1 = ∑ k 1 +k 2 =K k 1 ≥1, k 2 ≥1 w k 1 e 1 w k 2 e 1 2! ∂ 2 F(z, w 0 ) ∂ w 2 , (5.48) with D Ke 1 = P Ke 1 (∂ )+ 1 z Q Ke 1 (∂ )+ 1 z 2 R Ke 1 -∂ F(z, w 0 ) ∂ w , P Ke 1 (∂ ) = ∂ 2 -2K∂ + (K 2 -1)
. One deduces the conclusion of lemma 5.8 at the rank K by the arguments used previously. For the valuation, observe that val w k 1 e 1 w k 2 e 1 ≥ 2(k 1 -1) + 2(k 2 -1) when k 1 + k 2 = K, thus val w k 1 e 1 w k 2 e 1 ≥ 2(K -2). As a matter of fact, for every k ≥ 2,

w (k,0) (z) = b k z -2(K-1) (1 + O(z -1 )) with b 1 = 1, b k = 1 2(k-1)(k+1) ∑ k-1 p=1 b p b k-p , k ≥ 2,
which easily provides b k = k 12 k-1 by induction. The reader will easily check that the recursive relations (5.46) are true. (See also [24,1]).

Remark 5.7. Here again, we are not interested in the whole formal fundamental solutions of equations (5.47), (5.48), which incorporate the general solution (e -λ 1 kz z -τ 1 k ) -1 C 1 e -λ 1 z z -τ 1 w e 1 +C 2 e -λ 2 z z -τ 2 w e 2 of the associated homogeneous linear ODEs D (k,0) w = 0 (cf. lemma 5.4). Taking into account the term (• • • ) w e 1 would imply a rescaling of U 1 . The other term (• • • ) w e 2 concerns the other transseries.

Formal integral for the first Painlevé equation

We made general considerations on formal integrals in Sect. 5.2. We started the study of the formal integral for the prepared equation (3.6) associated with the first Painlevé equation in Sect. 5.3 : this gave us the transseries described by proposition 5.2. When no resonances occur, one gets with quite similar arguments the formal integral. However, this is not that simple for the first Painlevé equation where we have to cope with resonances.

Notations and preliminary results

Notations

It will be useful for our purpose to introduce the following notations: Definition 5.7. For any n ∈ N , we set n = n(1, 1) and

Ξ n,0 = {k = (k 1 , k 2 ) ∈ N 2 \ {0} | k 1 < n or k 2 < n} ∪ {n}.
We also set Ξ 0,0 = {(0, 0)}. We have already seen (proposition 5.2) that these properties have no consequence for the transseries but, as we shall see, this produces new phenomena when the formal integral is concerned, these being essentially consequences of the following fact, derived from lemma 5.3. Lemma 5.9. For every n ∈ N, k ∈ N 2 , the following identities are satisfied:

Example 5.3. Ξ 1,0 = (N × {0}) ∪ ({0} × N ) ∪ {(1, 1)}, Ξ 2,0 = (N × {0, 1}) ∪ ({0, 1} × N ) ∪ {(2, 2)}. Notice that for every n ∈ N, Ξ n+1,0 \ Ξ n,0 = n + Ξ 1,0 .
P n+k = P k , D n+k = z τ τ τ.n D k z -τ τ τ
.n , τ τ τ.n = -3n.

Preliminary lemmas

In a moment, we will have to deal with formal expansions of the type Lemma 5.10. We suppose n, N ∈ N and p ∈ N. We consider the ordinary differential equation

p ∑ l=0 log l (z) f l (z), p ∈ N, with the f l 's in C[[z -1 ]].
P(∂ )w + 1 z R(∂ )w = f (z), f (z) ∈ p l=0 log l (z)C[[z -1 ]], (5.49)
P(∂ ) = n ∑ m=0 α n-m ∂ m ∈ C[∂ ], α n = 0, R(∂ ) = n-1 ∑ m=0 γ n-m (z)∂ m ∈ C[[z -1 ]][∂ ]
Then (5.49) has a unique solution w ∈

(∂ )w + 1 z R(∂ )w = f p ; 2. if val f p < val ∑ p-1 l=0 log l (z) f l then val w p < val ∑ p-1 l=0 log l (z) w l .
Proof. One easily sees that the arguments used for the proof of lemma 5.5 can be extended, when observing that val ∂ ∑

l log l (z) f l ≤ val ∑ l log l (z) f l + 1.
We have seen in lemma 5.7 that the operators D e i , i = 1, 2, have specific behaviours. This is the purpose of the following lemma. Lemma 5.11. We suppose p ∈ N and i ∈ {1, 2}. We assume that f

= ∑ p l=0 log l (z) f l ∈ p l=0 log l (z)C[[z -1 ]] satisfies the conditions: 1. val f p = 1, f p = f p 1 z -1 (1 + 0(z -1 )), f p 1 = 0 2. val ∑ p-1 l=0 log l (z) f l ≥ 2.

Then the equation D

e i w = f has a unique solution w = ∑ p+1 l=0 log l (z) w l in p+1 l=0 log l (z)C[[z -1 ]] that satisfies the condition val ∑ p l=0 log l (z) w l ≥ 1. Moreover w p+1 = f p 1 (p + 1)P (-λ i ) w e i .
Otherwise, the general solution of the ODE D e i w = f in p+1 l=0 log l (z)C[[z -1 ]] is of the form w = w +U w e i where U ∈ C.

Proof. We examine the case i = 1 only. The ODE D e 1 w = f is equivalent to the equation :

P (-λ 1 )∂ w = f + -∂ 2 + 15 4 1 z 2 + ∂ F(z, w 0 ) ∂ w w, P (-λ 1 ) = -2.
By arguments already used in the proof of lemma 5.7, this problem amounts to looking for a formal solution which satisfies the fixed-point problem

w = U(z) + 1 P (-λ 1 ) ∂ -1 -∂ 2 + 15 4 1 z 2 + ∂ F(z, w 0 ) ∂ w w where U(z) = ∂ -1 f P (-λ 1 ) = f p 1 (p + 1)P (-λ 1 ) log p+1 (z) + p ∑ l=0 log l (z)O(z -1
).

Notice that we take the primitive with no constant term. This fixed-point problem has a unique formal solution under the form

w = f p 1 (p + 1)P (-λ 1 ) w e i log p+1 (z) + p ∑ l=0 log l (z) w l
and val ∑ p l=0 log l (z) w l ≥ 1. Eventually one can add to this particular solution any solution of the homogeneous equation D e i w = 0, that is any term of the form U w e i with U ∈ C.

Painlevé I: formal integral

We are now in position to detail the formal integral associated with the first Painlevé equation.

Theorem 5.1. We consider the ODE (3.6). Let be λ λ λ = (λ 1 , λ 2 ) = (1, -1) where the λ i 's are the multipliers, and

τ τ τ = (τ 1 , τ 2 ) = - 3 2 , - 3 2 , τ i = Q(-λ i ) P (-λ i ) , i = 1, 2. We set V k = U k e -λ λ λ .kz z -τ τ τ.k for any k ∈ N 2 and any U = (U 1 ,U 2 ) ∈ C 2 . We write n = n(1, 1) for any n ∈ N.
There exists a two-parameter formal solution of (3.6), freely depending on U ∈ C 2 , of the form

w(z, U) = w 0 (z) + ∞ ∑ n=0 ∑ k∈Ξ n+1,0 \Ξ n,0 V k w k (z), (5.50)
and uniquely determined by the following conditions:

1. w 0 ∈ C[[z -1 ]]; 2. w k = n ∑ l=0 log l (z) w [l] k ∈ n l=0 log l (z)C[[z -1 ]], for every k ∈ Ξ n+1,0 \ Ξ n,0 , n ∈ N;
3. for i = 1, 2, w e i satisfies w e i (z) = 1 + O(z -1 );

4. for every n ∈ N and i = 1, 2,

w n+e i = ∑ n l=0 log l (z) w [l] n+e i satisfies val w [n] n+e 1 < val ∑ n-1 l=0 log l (z) w [l] n+e i .
Moreover, the following properties are satisfied:

5. changing the normalization of w e i , i = 1, 2, is equivalent to rescaling the parameter U ∈ C 2 ;

6. for every n ∈ N and every

k ∈ Ξ n+1,0 \ Ξ n,0 , w k ∈ n l=0 log l (z)R[[z -1 ]]. Further- more w [l] (k 1 ,k 2 ) (z) = w [l]
(k 2 ,k 1 ) (-z) for every l ∈ [0, n]; 7. for every n ∈ N and every k ∈ Ξ n+1,0 \ Ξ n,0 ,

w k = n ∑ l=0 1 l! (κ κ κ.k) l z τ.l log l (z) w [0] k-l
(5.51)

where κ κ κ = (κ 1 , κ 2 ) = ( 5 12 , - 5 12
) is defined by:

κ i = a 2 P (-λ i ) 1 P(0) + 1 2! 1 P(-2λ i ) = 5 12 λ i , i = 1, 2, (5.52)
whereas a is given by

∂ 2 F(z, 0) ∂ w 2 = az -2 + o(z -2 ). As a consequence, for every n ∈ N , w n ∈ R[[z -1 ]] ; 8. for every k ∈ N 2 \ {0}, val w [0] k = 2(|k| -1).
Proof. Once for all:

• the property 5. is easily derived by an argument of homogeneity;

• the realness and eveness in property 6. are just consequences of the realness of equation (3.6) and its property of being be invariant under the change of variable z → -z, and to the chosen normalizations.

In what follows, we investigate the terms under the form w k with k ∈ Ξ n+1,0 \ Ξ n,0 and n ∈ N. We first look at what happens when n = 0 and n = 1, step by step so as to draw some conclusions, then we complete the proof by induction on n.

Case n = 0 and k = 1 This is the first case where a resonance appears. However, this case yields no surprise. Indeed, equation (5.37) for k = 1 reads

P 1 (∂ )w 1 + 1 z Q 1 (∂ )w 1 = - 1 z 2 R 1 + ∂ F(z, w 0 ) ∂ w w 1 + w e 1 w e 2 ∂ 2 F(z, w 0 ) ∂ w 2 (5.53) with P 1 (∂ ) = P 0 (∂ ) = ∂ 2 -1.
Therefore lemma 5.5 can be applied and one gets a unique solution

w 1 ∈ C[[z -1 ]] with, moreover, val w 1 = 2 and w 1 (z) = a P(0) z -2 + o(z -2 )
where a = 1 is given by: ∂ 2 F(z,0)

∂ w 2 = az -2 + o(z -2 ).
Explicit calculation yields:

w 1 (z) = -z -2 - 9 8 z -4 - 902139 80000 z -6 -• • •.
Cases n = 1 and k ∈ Ξ 2,0 \ Ξ 1,0

Cases k = 1 + e i , i = 1, 2 These are the first cases of semi-positive resonances and are more serious. Let us concentrate on the case k = 1 + e 1 for which equation (5.37) is

D 1+e 1 w 1+e 1 = ( w 1 w e 1 + w 2e 1 w e 2 ) ∂ 2 F(z, w 0 ) ∂ w 2 ,
that is also, from lemma 5.9 and proposition 5.2,

D e 1 (z 3 w 1+e 1 ) = g 1+e 1 ,
(5.54)

g 1+e 1 = z 3 ( w 1 w e 1 + w 2e 1 w e 2 ) ∂ 2 F(z, w 0 ) ∂ w 2 = 1 P(0) + 1 2! 1 P(-2λ 1 ) a 2 z -1 + O(z -2 ) = - 5 6 z -1 + O(z -2 ).
The conditions of application of lemma 5.11 are fulfilled: equation ( 5.54) has a oneparameter family of formal solutions, depending on U [1],1 ∈ C, of the form 5 12 .

w 1+e 1 = w 1+e 1 +U [1],1 z -3 w e 1 , w 1+e 1 = w [1] 1+e 1 log(z) + w [0] 1+e 1 , w [1] 1+e 1 = κ 1 z -3 w e 1 , val w [0] 1+e 1 ≥ 4. κ 1 = a 2 P (-λ 1 ) 1 P(0) + 1 2! 1 P(-2λ 1 ) =
(5.55) Explicitly, 1+e 1 ≥ 4 characterizes the particular solution w 1+e 1 among the one-parameter family of solutions.

w [0]
The case k = 1 + e i is deduced from the above result from the invariance of (3.6) under the change of variable z → -z. One gets a one-parameter family of formal solutions, depending on U [1],2 ∈ C, of the form

w 1+e 2 = w 1+e 2 +U [1],2 w e 2 , w 1+e 2 = w [1] 1+e 2 log(z) + w [0] 1+e 2 , w [1] 1+e 2 (z) = w [1] 1+e 1 (-z) = κ 2 z -3 w e 2 (z), w [0] 1+e 2 (z) = w [0] 1+e 1 (-z) κ 2 = a 2 P (-λ 2 ) 1 P(0) + 1 2! 1 P(-2λ 2 ) = -5
12 .

(5.56)

In the sequel, we fix

U [1],1 = U [1]
,2 = 0, that is we only consider the (well and uniquely defined) particular solutions w 1+e i , i = 1, 2.

We stress that adding terms of the form U [1],1 w e 1 and U [1],2 w e 2 has the effect to rescaling the parameter (U 1 ,U 2 ). In particular, changing the branch of the log has non consequence for the formal integral.

Cases k = 1 + ke i One step further, we consider the case k = 1 + 2e i . We take i = 1 only for simplicity. From (5.37) and lemma 5.9, we get:

(5.57)

D 2e 1 (z 3 w 1+2e 1 ) = z 3 w 1+e 1 w e 1 + w 2e 1 w 1 + w 3e 1 w e 2 ∂ 2 F(z, w 0 ) ∂ w 2 .
By proposition 5.2 and the above result, the right-hand side of equation (5.57) is a formal series expansion of the type f = f [1] log(z) + f [0] with val f [1] = 2 and val f [0] = 3. Applying lemma 5.10, we get for (5.57) a unique formal solution of the form w 1+2e 1 = w

[1] 1+2e 1 log(z) + w [0] 1+2e 1 ∈ 1 l=0 log l (z)C[[z -1 ]] with val w [1]
1+2e 1 = 5 and val w

[0] 1+2e 1 = 6. Moreover, w [1] 1+2e 1 solves the ODE D 2e 1 (z 3 w [1] 1+2e 1 ) = z 3 w [1] 1+e 1 w e 1 ∂ 2 F(z, w 0 ) ∂ w 2 = κ 1 w 2 e 1 ∂ 2 F(z, w 0 ) ∂ w 2 .
Comparing to (5.47), one concludes that

w 1+2e 1 = w [1] 1+2e 1 log(z) + w [0] 1+2e 1 , w [1] 1+2e 1 = 2κ 1 z -3 w 2e 1 , val w [0] 1+2e 1 = 6.
We now reason by induction, assuming that for every k ∈ [2, K -1] with K ≥ 3, one has

w 1+ke 1 = w [1] 1+ke 1 log(z) + w [0] 1+ke 1 , w [1] 1+ke 1 = kκ 1 z -3 w ke 1 , val w [0] 1+ke 1 = 2(k + 1)
. Then, by (5.37) and lemma 5.9,

D Ke 1 (z 3 w 1+Ke 1 ) = z 3 ∑ k 1 +k 2 =1+Ke 1 |k 1 |≥1, |k 2 |≥1 w k 1 w k 2 2 ∂ 2 F(z, w 0 ) ∂ w 2 (5.58) = z 3 ∑ k 1 +k 2 =K k 1 ≥1, k 2 ≥1 w 1+k 1 e 1 w k 2 e 1 ∂ 2 F(z, w 0 ) ∂ w 2 + z 3 w 1 w Ke 1 + w (1+K)e 1 w e 2 ∂ 2 F(z, w 0 ) ∂ w 2
With the above reasoning, one gets a unique solution

w 1+Ke 1 = w [1] 1+Ke 1 log(z) + w [0] 1+Ke 1 ∈ 1 l=0 log l (z)C[[z -1 ]] where w [1]
1+Ke 1 solves the ODE

D Ke 1 (z 3 w [1] 1+Ke 1 ) = κ 1 ∑ k 1 +k 2 =K k 1 ≥1, k 2 ≥1 k 1 w k 1 e 1 w k 2 e 1 ∂ 2 F(z, w 0 ) ∂ w 2 = Kκ 1 ∑ k 1 +k 2 =K k 1 ≥1, k 2 ≥1 w k 1 e 1 w k 2 e 1 2 ∂ 2 F(z, w 0 ) ∂ w 2
Comparing to (5.48), one concludes that

w 1+Ke 1 = w [1] 1+Ke 1 log(z) + w [0] 1+Ke 1 , w [1] 1+Ke 1 = Kκ 1 z -3 w Ke 1 , val w [0] 1+Ke 1 = 2(K + 1).
Case k = (2, 2) What remains to do when k ∈ Ξ 2,0 \ Ξ 1,0 is to examine the case k = (2, 2). By (5.37) and lemma 5.9,

D 1 (z 3 w 2 ) = z 3 w 1+e 1 w e 2 + w 1+e 2 w e 1 + w 2e 1 w 2e 2 + 1 2 w 1 w 1 ∂ 2 F(z, w 0 ) ∂ w 2 .
(5.59)

We observe from (5.55) and (5.56) that

w [1] 1+e 1 w e 2 + w [1]
1+e 2 w e 1 = κ 1 z -3 w e 1 w e 2 + κ 2 z -3 w e 2 w e 1 = 0.

Therefore the log-term disappears in the right-hand side of (5.59) as a consequence of the symmetries of the problem. Moreover val ( w

[0] 1+e 1 w e 2 + w [0] 1+e 2 w e 1 + w 2e 1 w 2e 2 + 1 2 w 1 w 1 ) ≥ 4.
By lemma 5.10, we get

w 2 ∈ C[[z -1 ]
] with val w 2 = 6. Explicit calculation provides:

w 2 (z) = - 5 6 1 z 6 - 2177 432 1 z 8 - 5288521 54000 1 z 10 + • • • . Induction
We assume that N is an integer ≥ 2 and we suppose that the properties announced in theorem 5.1 are true for any integer n ∈ [0, N -1] and any k ∈ Ξ n+1,0 \ Ξ n,0 .

We notice on the one hand that Ξ N+1,0 \ Ξ N,0 = 1 + Ξ N,0 \ Ξ N-1,0 . On the other hand, for every

k ∈ Ξ N,0 \ Ξ N-1,0 , D 1+k ( w 1+k ) = ∑ k 1 +k 2 =1+k |k 1 |≥1, |k 2 |≥1 w k 1 w k 2 2 ∂ 2 F(z, w 0 ) ∂ w 2 (5.60)
We set X = log(z) and we consider X as an indeterminate. The right-hand side of (5.60) is of the form

f = ∑ f [l] X l with ∂ X f = ∂ X ∑ k 1 +k 2 =1+k |k 1 |≥1, |k 2 |≥1 w k 1 w k 2 2 ∂ 2 F(z, w 0 ) ∂ w 2 = ∑ k 1 +k 2 =1+k |k 1 |≥1, |k 2 |≥1 (∂ X w k 1 ) w k 2 + w k 1 (∂ X w k 2 ) 2 ∂ 2 F(z, w 0 ) ∂ w 2 .
Using the induction hypothesis, when

1 + k 1 ∈ Ξ n+1,0 \ Ξ n,0 , for any n ∈ [0, N -1], ∂ X n ∑ l=1 w [l] 1+k 1 X l = (κ κ κ.k 1 )z -3 n-1 ∑ l=0 w [l] k 1 X l , that is ∂ X w 1+k 1 = (κ κ κ.k 1 )z -3 w k 1 . Therefore: ∂ X f = z -3 ∑ k 1 +k 2 =k |k 1 |≥1, |k 2 |≥1 (κ κ κ.k 1 ) w k 1 w k 2 ∂ 2 F(z, w 0 ) ∂ w 2 = (κ κ κ.k)z -3 ∑ k 1 +k 2 =k |k 1 |≥1, |k 2 |≥1 w k 1 w k 2 2 ∂ 2 F(z, w 0 ) ∂ w 2 .
Thus ∂ X f = (κ κ κ.k)z -3 D k ( w k ) and (5.60) provides:

∂ X D k (z 3 w 1+k ) = (κ κ κ.k)D k ( w k ).
Observing that ∂ X D k ∂ -1 X = D k , one easily gets w 1+k either from lemma 5.11 or lemma 5.10, with w 1+k = (κ κ κ.k)z -3 ∂ -1 X w k . The property for w n+1 is easy and is left to the reader. This ends the proof of theorem 5.1.

Definition 5.9. The two-parameter formal solution defined by theorem 5.1 is the formal integral of the prepared ODE (3.6) associated with the first Painlevé equation. The coefficients λ i , τ i and κ i , i = 1, 2, are the formal invariants. The formal series w

[0]
k are called the k-th series of the formal integral. We set

W [0] k = z -τ τ τ.k w [0] k and W k = z -τ τ τ.k w k for any k ∈ N 2 .
Remark 5.8. Theorem 5.1 can be compared to [24] and specially to [1], where the calculations made there translate into ours up to renormalization. We also mention obvious links between Theorem 5.1 and the instanton-type solutions of Kawai et al [28,2]. Definition 5.10. For any k ∈ N 2 , one denotes by E k and F k the following operators:

E k = κ κ κ.k z 4 P (∂ -λ λ λ .k) + κ κ κ.k z 5 Q (∂ -λ λ λ .k) - τ.(2k -1) + 1 2! P (∂ -λ λ λ .k) = 2 κ κ κ.k z 4 (∂ -λ λ λ .k) - κ κ κ.k z 5 (τ.(2k -1) + 4) , F k = 1 2! (κ κ κ.k) 2 z 8 P (∂ -λ λ λ .k) = (κ κ κ.k) 2 z 8 .
We need hardly mention the analogue of lemma 5.9.

Lemma 5.12. For every n ∈ N, k ∈ N 2 ,

E n+k = z τ τ τ.n E k z -τ τ τ.n , F n+k = z τ τ τ.n F k z -τ τ τ.n .
We finally give a corollary stemming from theorem 5.1.

Corollary 5.1. The formal integral (5.50) associated with the prepared ODE (3.6) can be written under the form:

w(z, U) = ∑ k∈N 2 V k w [0] k , V k = U k e -(λ λ λ .k)z+(κ κ κ.k)U 1 log(z) z -τ τ τ.k . (5.61) Equivalently, w(z, U) = Φ(z,U 1 e -λ 1 z-(τ 1 -κ 1 U 1 ) log(z) ,U 2 e -λ 2 z-(τ 2 -κ 2 U 1 ) log(z) )
where

Φ(z, u) = ∑ k∈N 2 u k w [0] k (z) ∈ C[[z -1 , u]
] is solution of the equation:

P ∂ z - 2 ∑ i=1 (λ i + τ i -κ i u 1 z )u i ∂ u i Φ + 1 z Q ∂ z - 2 ∑ i=1 (λ i + τ i -κ i u 1 z )u i ∂ u i Φ = F(z, Φ).
(5.62) The formal series w

[0] k ∈ z -2|k|+2 R[[z -1 ]] satisfy: • for any k ∈ Ξ 1,0 \ Ξ 0,0 , D k w [0] k = ∑ k 1 +k 2 =k |k i |≥1 w [0] k 1 w [0] k 2 2! ∂ 2 F(z, w 0 ) ∂ w 2 ; • for any k ∈ Ξ 2,0 \ Ξ 1,0 , D k w [0] k + E k w [0] k-1 = ∑ k 1 +k 2 =k |k i |≥1 w [0] k 1 w [0] k 2 2! ∂ 2 F(z, w 0 ) ∂ w 2 ; • otherwise, D k w [0] k + E k w [0] k-1 + F k w [0] k-2 = ∑ k 1 +k 2 =k |k i |≥1 w [0] k 1 w [0] k 2 2! ∂ 2 F(z, w 0 ) ∂ w 2 ;
Proof. Let us examine (5.50) more closely. The formal integral can be written as follows:

w(z, U) = ∞ ∑ n=0 V n w n (z) + ∑ i=1,2 ∞ ∑ k=1 ∞ ∑ n=0 V n+ke i w n+ke i (z), (5.63)
that is we consider the sums along the direction given by the vector (1, 1) that determines the resonance. We set

T k = U k e -(λ λ λ .k)z+(κ κ κ.k)U 1 log(z) z -τ τ τ.k .
For the first sum we know that each

w n (z) belongs to C[[z -1 ]] and ∑ ∞ n=0 V n w n = ∑ ∞ n=0
T n w n because κ.n = 0. We now look at the other sums and we use the relations given by (5.51). We get for

i = 1, 2, ∞ ∑ k=1 ∞ ∑ n=0 V n+ke i w n+ke i = ∞ ∑ k=1 V ke i ∞ ∑ n=0 V n n ∑ l=0 1 l! κ i kz -3 log(z) l w [0] n-l+ke i = ∞ ∑ n=0 V n ∞ ∑ k=1 V ke i e (κikU 1 log(z)) w [0] n+ke i . = ∞ ∑ n=0 ∞ ∑ k=1 T n+ke i w [0] n+ke i .
The equation (5.62) is obtained by the arguments developed in remark 5.3. The reader will check that equation (5.62) is equivalent to the given hierarchy of equations.

Let us write u 1 (z) = U 1 e -λ 1 z-(τ 1 -κ 1 U 1 ) log(z) , u 2 (z) = U 2 e -λ 2 z-(τ 2 -κ 2 U 1 ) log(z) and observe that t (u 1 , u 2 ) provides the general analytic solution for a non linear differential equation that only depends on the formal invariants:

∂ u 1 u 2 + λ 1 + τ 1 z 0 0 λ 2 + τ 2 z u 1 u 2 = κ 1 z 4 u 1 u 2 0 0 κ 2 z 4 u 1 u 2 u 1 u 2 .
(5.64)

This means that corollary 5.1 can be written in term of formal classification and of (canonical) normal form:

Corollary 5.2. There exists a formal transformation w = Φ(z, u) of the form

Φ(z, u) = ∑ k∈N 2 u k w [0] k (z), w [0] k ∈ C[[z -1 ]],
(5.65) that formally transforms the prepared ODE (3.6) into the normal form equation:

∂ u + B 0 (z)u = B 1 z, u u (5.66) B 0 = λ 1 + τ 1 z 0 0 λ 2 + τ 2 z , B 1 (z, u) = u 1 z 4 κ 1 0 0 κ 2 , u 1 = u 1 u 2 .

Comments

Analogues of proposition 5.1 can be stated for differential equations, resp. difference equations, of order 1 and dimension n, with one level and no resonance, given in prepared form :

∂ v + B 0 (z)v = g(z, v) (5.67) with B 0 (z) = j λ j I n j + z -1 M j , ∑ j n j = n, resp. v(z + 1) = B 0 (z)v(z) + g(z, v) (5.68)
with B 0 (z) = j e -λ j z (1+z -1 ) M j . In each case, there exists a formal transformation

of the type v = T (z, u), T (z, u) = ∑ k∈N n u k v k (z), v k (z) ∈ C n [[z -1 ]
] that brings the equation to the linear normal form ∂ u + B 0 (z)u = 0, resp. u(z + 1) = B 0 (z)u(z).

To be correct, the upshot for difference equations is more subtle.

This property is still valid for differential equations with more than one level, see [31,5,8] and references therein. In particular, the whole set of formal invariants is already given by the linear part (in Jordan form) of the equation.

When resonances occur and as we saw with the first Painlevé equation, the normal form equation is nonlinear and incorporates new formal invariants. This is essentially a consequence of the Poincaré-Dulac theorem [3]; for instance in (5.66), one recognizes the effect of the positively resonance of order 3 with the resonances monomials u 2 1 u 2 and u 1 u 2 2 . The classification is detailed in [19], see also [21] where the notion of (so-called) moulds and arborification are used (a good introduction of which is [37]).

Chapter 6

Truncated solutions for the first Painlevé equation Abstract In the previous chapters, we studied the unique formal solution of the first Painlevé equation then we introduced its formal integral. In this chapter, we show that formal series components of the formal integral are 1-Gevrey and their minors have analytic properties quite similar to those for the minor of the formal series solution we started with (Sect. 6.1). We then make a focus on the transseries solution and we show their Borel-Laplace summability (Sect. 6.2). This provides the truncated solutions by Borel-Laplace summation (Sect. 6.4).

Borel-Laplace summability of the k-th series and beyond

We described with theorem 5.1 and its corollary 5.1 the formal integral

w(z, U) = ∑ k∈N 2 V k w [0]
k associated with the first Painlevé equation. Our goal in this section is mainly to show the following assertion. Theorem 6.1. For every k ∈ N 2 , the k-th series w

[0]
k is 1-Gevrey, its minor w k can be analytically continued to the Riemann surface R (1) , with at most exponential growth of order 1 at infinity on R (1) .

We already know by theorem 3.3 and theorem 4.2 that w 0 = w [0] 0 enjoyes the above properties. Our task comes down to studying the other k-th series. This is what we do in what follows and we start with some preliminaries.

Preliminary results

In what follows we use a notation introduced in definition 5.5. Lemma 6.1. We set P(∂ ) = ∂ 2 -1 and for every k ∈ N 2 , P k (∂ ) = P(-λ λ λ .k + ∂ ) with λ λ λ = (λ 1 , λ 2 ) = (1, -1). For i = 1, 2, we denote by Pe i (∂ ) the operator defined by P e i (∂ ) = Pe i (∂ )∂ so that Pe i (-λ i ) = 0. Then, for any ρ ∈]0, 1[, there exists M ρ,(0) > 0 such that, for every ζ ∈ C \ m∈Z D(m, mρ) :

1. for i = 1, 2, 1 Pe i (-ζ ) ≤ M ρ,(0) ; 113 2. for every k ∈ Ξ 1,0 with |k| ≥ 2, for m = 0, 1, (ζ + λ λ λ .k) m P k (-ζ ) ≤ M ρ,(0) |k| -1 and, for k = (1, 1), 1 P k (-ζ ) ≤ M 2 ρ,(0) |k| 2 -1 .
Moreover one can choose M ρ,(0) = 1 ρ .

Proof. We only examine the case k ∈ Ξ 1,0 \ {(1, 1)} with |k| > 1. With no loss of generality, we can assume that k = (k, 0) with k ≥ 2. Thus

P k (-ζ ) = (ζ + k -1)(ζ + k + 1), ζ + λ λ λ .k = ζ + k and we notice that |ζ + k -1| ≥ (k -1)ρ and |ζ + k + 1| ≥ (k + 1)ρ for ζ ∈ C \ m∈Z D(m, mρ). Therefore, 1 |P k (-ζ )| ≤ 1 (k 2 -1)ρ 2 for ζ ∈ C\ m∈Z D(m, mρ). Now either ℜ(ζ + k) ≥ 0, thus |ζ + k + 1| ≥ max{1, |ζ + k|} and therefore max{1,|ζ +λ λ λ .k|} |P k (-ζ )| ≤ 1 (k-1)ρ ; or else ℜ(ζ + k) ≤ 0, which implies |ζ + k -1| ≥ max{1, |ζ + k|} and finally max{1,|ζ +λ λ λ .k|} |P k (-ζ )| ≤ 1 (k+1)ρ .
Lemma 6.2. We follow the conditions of lemma 6.1. We set Q(∂ ) = -3∂ , while Q k (∂ ), R k (∂ ) are given by (5.31), (5.32) with τ = -3 2 , -3 2 . Then, for every

k ∈ Ξ 1,0 \ {(1, 1)} with |k| > 1, for every ζ ∈ • R (0) ρ , |Q k |(|ζ |) |P k (-ζ )| ≤ 3M ρ,(0) , |R k |(|ζ |) |P k (-ζ )| ≤ 9 4 M 2 ρ,(0) .
Proof. We notice that lemma 6.1 can be applied for ζ ∈

The e i -th series

We start our proof of theorem 6.1 by paying special attention to w e i = w ρ . Moreover, w e i can be analytically continued to the Riemann surface R (1) , with at most exponential growth of order 1 at infinity on R (1) .

Proof. It is enough to study w e 1 since w e 2 (z) = w e 1 (-z). We know that w e 1 solves (5.45), namely:

∂ Pe 1 (∂ ) w e 1 = 15 4 1 z 2 + ∂ F(z, w 0 ) ∂ w w e 1 , Pe i = ∂ -2. (6.1)
The formal Borel transform of w e 1 reads B( w e 1 ) = δ + w e 1 where the minor

w e 1 (ζ ) ∈ C[[ζ ]
] satisfies the following convolution equation, deduced from (6.1):

∂ Pe 1 ( ∂ ) w e 1 = 15 4 ζ + ∂ F(ζ , w 0 ) ∂ w * (δ + w e 1 ). (6.2)
In this equation, we use the notation:

∂ F(ζ , w 0 ) ∂ w = f 1 (ζ ) + 2 f 2 * w 0 (ζ ) = -4ζ + ζ * w 0 (ζ ). (6.3)
Equation ( 6.2) can be thought of as a linear differential equation with a regular singular point at 0.

Instead of (6.2), consider the convolution equation

∂ Pe 1 ( ∂ ) w = a 1 ζ + a 2 ζ 2 2! * (δ + w). Set g = ∂ Pe 1 ( ∂ ) w = ζ (ζ + 2) w. For ζ = 0, one gets g = a 1 ζ + a 2 ζ 2 2! * δ + g ζ (ζ + 2) .
This implies by differentiation that g (4) = a 1 g

ζ (ζ + 2) (2)
+ a 2 g

ζ (ζ + 2) (1)
where

g (i) = d i g dζ i .
The last ODE has a regular singular point at 0. One can apply the same trick to (6.2) but for the fact of getting an infinite order differential operator. Equation (6.2) can be analyzed with the tools developed in Sect. 3.3.2. We introduce

G(ζ ) = 15 4 ζ + ∂ F(ζ , w 0 ) ∂ w = - ζ 4 + ζ * w 0 (ζ )
and we remark that G belongs to the

maximal ideal M O( • R (0) ρ ) of O( • R (0) ρ ) for any ρ ∈]0, 1[, thus ∂ -1 G ∈ O( • R (0)
ρ ) is well-defined. We set w e 1 = P-1 e 1 ( ∂ ) ∂ -1 G + v e 1 and (6.2) becomes

∂ Pe 1 ( ∂ ) v e 1 = G * P-1 e 1 ( ∂ ) ∂ -1 G + G * v e 1 . (6.4)
Observe that G * P-1

e 1 ( ∂ ) ∂ -1 G belongs to M O( • R (0) ρ ). Let R > 0 be any real positive number, U R be the star-shaped domain U R = D(0, R) ∩ • R (0)
ρ and we set B r = { v ∈ O(U R ), v ν ≤ r}, for r > 0 and ν > 0. By proposition 3.9 and lemma 6.1, when ν → ∞,

P-1 e 1 ( ∂ ) ∂ -1 G * P-1 e 1 ( ∂ ) ∂ -1 G ν → 0. Explicitly P-1 e 1 ( ∂ ) ∂ -1 G * P-1 e 1 ( ∂ ) ∂ -1 G ν ≤ M ρ,(0) R ∂ -1 G * P-1 e 1 ( ∂ ) ∂ -1 G ν ≤ M ρ,(0) νR 2 ∂ -1 G 0 P-1 e 1 ( ∂ ) ∂ -1 G ν .
Also, P-1 The same reasoning can be applied for showing that w e 1 can be analytically continued to R (1) , in application of proposition 4.5 and theorem 4.2.

e 1 ( ∂ ) ∂ -1 G * v e 1 ν ≤ M ρ,(0) νR 2 ∂ -1 G 0 v e 1 ν ,
To get upper bounds, we notice by (6.3) and lemma 3.3 that for every ζ ∈ This means that w e 1 has an analytic Laplace transform under the form 1 :

• R (0) ρ , ∂ -1 G(ζ ) ≤ 1
w e 1 (z) = ∑ n≥1 1 ρ n 1 4z + 1 z A z -τ n , A = 4.22, τ = 4.22 ρ .
When assuming |z| ≥ 5.81 ρ , for instance, one gets 1

ρ 1 4z + 1 z A z -τ ≤ 0.5 (since ρ < 1), thus | w e 1 (z)| ≤ 1.
Therefore by lemma 3.5, for any 0 < ρ < 1, for every

ζ ∈ • R (0) ρ , | w e 1 (ζ )| ≤ 5.81 ρ e 5.81
ρ |ξ | . One shows in the same way that w e 1 has at most exponential growth of order 1 at infinity on R (1) , using lemma 4.11 and theorem 4.2.

The ke i -th series

We now turn to the ke i -th series, that is the terms w ke i = w

[0] ke i of the transseries, for k ≥ 2. Lemma 6.4. For every integer k ≥ 2, the k-th series w ke

i ∈ z -2(k-1) C[[z -1 ]] is 1-

Gevrey, its minor w ke i defines a holomorphic function on

• R (0) with at most exponential growth of order 1 at infinity. Moreover, w ke i can be analytically continued to the Riemann surface R (1) , with at most exponential growth of order 1 at infinity on R (1) .

Proof. Once again from the invariance of the equation (3.6) under the symmetry z → -z, there is no loss of generality in studying only the ke i -th series w ke 1 .

We know that w 0 , w e 1 are holomorphic on • R (0) and can be analytically continued to R (1) . Moreover, for every ζ ∈

• R (0) ρ , | w 0 (ζ )| ≤ w 0 (ξ ), | w e 1 (ζ )| ≤ w e 1 (ξ ), ξ = |ζ | and for every ζ ∈ R (1) ρ , | w 0 (ζ )| ≤ w 0 (ξ ), | w e 1 (ζ )| ≤ w e 1 (ξ ), ξ = leng(ζ )
, where w 0 and w e 1 are entire functions, real positive and non-decreasing on R + , with at most exponential growth of order 1 at infinity. We know from lemma 5.8 and (5.48) that for every k ≥ 2,

w ke 1 (z) = ∑ l≥0 a ke 1 ,l z -l ∈ z -2(k-1) C[[z -1 ]] 1 We recall that B A z -τ = Ae τξ .
solves the differential equation

D ke 1 w ke 1 = ∑ k 1 +k 2 =k k 1 ≥1, k 2 ≥1 w k 1 e 1 w k 2 e 1 2! ∂ 2 F(z, w 0 ) ∂ w 2 . (6.6)
We deduce that the formal Borel transform B( w ke 1 ) = a ke 1 ,0 δ + w ke 1 has its minor which satisfies the identity2 :

D ke 1 w ke 1 = ∑ k 1 +k 2 =k k 1 ≥1, k 2 ≥1 (a k 1 e 1 ,0 δ + w k 1 e 1 ) * (a k 2 e 1 ,0 δ + w k 2 e 1 ) 2! * ∂ 2 F(ζ , w 0 ) ∂ w 2 (6.7)
where

∂ 2 F(ζ , w 0 ) ∂ w 2 = 2 f 2 (ζ ) = ζ , whereas D ke 1 w ke 1 = P ke 1 ( ∂ ) w ke 1 + 1 * Q ke 1 ( ∂ ) w ke 1 + ζ R ke 1 - ∂ F(ζ , w 0 ) ∂ w * w ke 1 (6.8) with ∂ F(ζ , w 0 ) ∂ w
given by (6.3). These equations (6.7) can be seen as linear differential equations with a regular point at 0. They are all of the type

p( ∂ ) w + 1 * [q( ∂ ) w] = ζ * [r( ∂ ) w] + N ∑ n=0 f n * w * n
(6.9) investigated in Sect. 3.3.2 and Sect. 4.5. We use the methods introduced there and make a proof by induction on k, considering the operators N k defined as follows:

N k v = P -1 (k,0) ( ∂ ) -1 * Q (k,0) ( ∂ ) v + -ζ R (k,0) + ∂ F(ζ , w 0 ) ∂ w * v + ∑ k 1 +k 2 =K k 1 ≥1, k 2 ≥1 (a k 1 e 1 ,0 δ + w k 1 e 1 ) * (a k 2 e 1 ,0 δ + w k 2 e 1 ) 2! * ∂ 2 F(ζ , w 0 ) ∂ w 2    .
Case k = 2 Let R > 0 be a real positive number, ρ ∈]0, 1[ and U R be the star-shaped

domain U R = D(0, R) ∩ • R (0) ρ . We set B r = { v ∈ O(U R ), v ν ≤
r} for r > 0 and ν > 0, and we look at the mapping

N 2 : v ∈ B r → N 2 v. We know that w (1,0) ∈ O( • R (0) ) while ∂ F(ζ , w 0 ) ∂ w and ∂ 2 F(ζ , w 0 ) ∂ w 2 belong to M O( • R (0)
ρ ). Using lemma 6.1 and arguments already used in Sect. 3.3.2.3, one easily shows that N 2 is a contractive map. Thus equation (6.7), k = 2 has a unique solution in B r . This shows, by uniqueness, that w 2e 1 can be continued holomorphically on • R (0) . When replacing U R by the open set of L-points U = U ρ,L ⊂ R (1) and arguing like what have been done for the proof of theorem 4.2, one shows that w 2e 1 can be holo-morphically continued to the Riemann surface R (1) .

To get upper bounds, we notice that for every ζ ∈

• R (0) ρ , ∂ F(ζ , w 0 ) ∂ w ≤ ∂ F ∂ w (ξ , w 0 ) and ∂ 2 F(ζ , w 0 ) ∂ w 2 ≤ ∂ 2 F ∂ w 2 (ξ , w 0 ) with ξ = |ζ |, ∂ 2 F ∂ w 2 (ξ , w 0 ) = 2| f 2 |(ξ ) = ξ and ∂ F ∂ w (ξ , w 0 ) = | f 1 |(ξ ) + 2| f 2 | * w 0 (ξ ) = 4ξ + ξ * w 0 (ξ ).
Using lemma 6.2 and the Grönwall lemma 3.9, we sees that for every ζ ∈

• R (0) ρ , | w 2e 1 (ζ )| ≤ w 2e 1 (ξ )
where w 2e 1 is the entire function, real positive on R + , with at most exponential growth of order 1 at infinity, satisfying the linear equation:

1 M ρ,(0) w 2e 1 = 3 + 9 4 M ρ,(0) ξ + ∂ F ∂ w (ξ , w 0 ) * w 2e 1 (6.10) + (δ + w e 1 ) * 2 2! * ∂ 2 F ∂ w 2 (ξ , w 0 ).
When working on R (1) , one rather argues with the Grönwall lemma 4.11, thus get-

ting | w 2e 1 (ζ )| ≤ w 2e 1 (ξ ) for every ζ ∈ R (1)
ρ . In these estimates, ξ = leng(ζ ), and w 2e 1 is the entire function, real positive and non-decreasing on R + , with at most exponential growth of order 1 at infinity, satisfying the linear equation:

1 M ρ,(1)
w 2e 1 = 3 + 9 4 M ρ,(1) ξ + ∂ F ∂ w (ξ , w 0 ) * w 2e 1 (6.11) + (δ + w e 1 ) * 2 2! * ∂ 2 F ∂ w 2 (ξ , w 0 ).
Induction Let K ≥ 3 be an integer greater than 3. We assume that for every integer 0) and can be analytically continued to R (1) . Furthermore,

k ∈ [0, K[, w ke 1 is holomorphic on • R (
for every ζ ∈ • R (0) ρ , | w ke 1 (ζ )| ≤ w ke 1 (ξ ), ξ = |ζ |, for every ζ ∈ R (1) ρ , | w ke 1 (ζ )| ≤ w ke 1 (ξ ), ξ = leng(ζ ),
where, in each case, w ke 1 is an entire function, real positive and non-decreasing on R + , with at most exponential growth of order 1 at infinity. 0) and can by analytically continued to R (1) .

One easily shows that the mapping N

K : v ∈ B r → N K v is a contractive, either working in (O(U R ), . ν ) or in (O(U ρ,L ), . ν ). Thus, by uniqueness, w Ke 1 is holo- morphic on • R (

We get upper bounds, either in

• R (0)
ρ with the Grönwall lemma 3.9, or in R

w Ke 1 = 3 + 9 4 M ρ,(0) ξ + ∂ F ∂ w (ξ , w 0 ) * w Ke 1 (6.12) + ∑ k 1 +k 2 =K k 1 ≥1, k 2 ≥1 (a k 1 e 1 ,0 δ + w k 1 e 1 ) * (a k 2 e 1 ,0 δ + w k 2 e 1 ) 2! * ∂ 2 F ∂ w 2 (ξ , w 0 ).
Also, for every ζ ∈ R

w Ke 1 = 3 + 9 4 M ρ,(1) ξ + ∂ F ∂ w (ξ , w 0 ) * w Ke 1 + ∑ k 1 +k 2 =K k 1 ≥1, k 2 ≥1 (a k 1 e 1 ,0 δ + w k 1 e 1 ) * (a k 2 e 1 ,0 δ + w k 2 e 1 ) 2! * ∂ 2 F ∂ w 2 (ξ , w 0 ).
This ends the proof of lemma 6.4.

The other k-th series

Looking at (5.53), one easily see that the above methods can be applied to study the minor w 1 = w

[0]

1 of the (1, 1)-series w 1 . Thus, theorem 6.1 is shown for k = 0 any k ∈ Ξ n+1,0 \ Ξ n,0 and with n = 1. The rest of the proof is made by induction on n, using the hierarchy of equations given in corollary 5.1 and the reasoning made above. This part holds no surprise and is left to the reader. This ends the proof of theorem 6.1.

Borel-Laplace summability of the transseries

We now restrict ourself to the transseries solution of the ODE (3.6), having in view of analyzing their Borel-Laplace summability. From the invariance of the equation (3.6) under the symmetry z → -z, it is enough to only focus on the transseries (5.41) associated with the multiplier λ 1 = 1, namely:

w(z,Ue 1 ) = ∞ ∑ k=0 V k w ke 1 (z), V k = U k e -λ 1 kz z -τ 1 k .
(6.13)

A useful supplement

We complete lemma 6.4 with the following result.

Lemma 6.5. For every ρ ∈]0, 1[, there exist A = A(ρ) > 0, τ = τ(ρ) > 0 and a sequence w ke 1 k≥2 of entire functions, real positive on R + , with the following properties:

• for every integer k ≥ 2, w ke 1 (ξ ) ∈ ξ 2k-3 C{ξ }; • for every ξ ∈ C, | w ke 1 (ξ )| ≤ 3 √ ρ 2 k Ae τ|ξ | , and for every integer m ∈ [1, 2k -3], | w ke 1 (ξ )| ≤ 3 √ ρ 2 k A m+1 ζ m-1 (m-1)! * e τζ (|ξ |). • for every ζ ∈ • R (0) ρ , | w ke 1 (ζ )| ≤ w ke 1 (ξ ) with ξ = |ζ |.

Moreover one can choose

A = τ = 27 4ρ
in the above estimates.

Proof. We know by theorem 3.3, lemma 6.3 and lemma 6.4 that, for every integer

k ∈ N, w ke 1 is holomorphic on • R (0) . Also, for every ρ ∈]0, 1[, for every ζ ∈ • R (0) ρ , | w ke 1 (ζ )| ≤ w ke 1 (ξ ) with ξ = |ζ |
where w 0 (ξ ) = A 0 e τ 0 ξ and w e 1 (ξ ) = A e 1 e τ e 1 ξ are convenient majorant functions while, for any integer k ≥ 2, w ke 1 solves the convolution equation (6.12). One first shows that for any integer k ≥ 2, w ke 1 (ξ ) belongs to ξ 2k-3 C{ξ } and we reason by induction: using the fact that

∂ 2 F ∂ w 2 (ξ , w 0 ) = O(ξ ), one sees that (δ + w e 1 ) * 2 * ∂ 2 F ∂ w 2 (ξ , w 0 ) = O(ξ ), thus w 2e 1 (ζ ) = O(ζ ); then,
by an induction hypothesis, we check that integer k ≥ 3 of the form

k = k 1 + k 2 with k 1 , k 2 ∈ N , (a k 1 e 1 ,0 δ + w k 1 e 1 ) * (a k 2 e 1 ,0 δ + w k 2 e 1 ) = O(ξ 2k-5
) (we recall that a ke 1 ,0 = 0 apart from a e 1 ,0 = 1), thus w ke 1 (ζ ) = O(ξ 2k-3 ) by (6.12).

We then introduce the generating function w(ξ,V)

= ∞ ∑ k=2
V k w ke 1 (ξ ) and we deduce from (6.12) that w satisfies the identity:

1 M ρ,(0) w = 3 + 9 4 M ρ,(0) ξ + ∂ F ∂ w (ξ , w 0 ) * w + ∞ ∑ k=2 V k ∑ k 1 +k 2 =k k 1 ≥1, k 2 ≥1 (a k 1 e 1 ,0 δ + w k 1 e 1 ) * (a k 2 e 1 ,0 δ + w k 2 e 1 ) 2! * ∂ 2 F ∂ w 2 (ξ , w 0 ).
This can be written also as follows (remember: a ke 1 ,0 = 0 apart from a e 1 ,0 = 1):

1 M ρ,(0) w = 3 + 9 4 M ρ,(0) ξ + ∂ F ∂ w (ξ , w 0 ) * w+ V δ + w e 1 + w * 2 2! * ∂ 2 F ∂ w 2 (ξ , w 0 ).
Explicitly, one can choose M ρ,(0) = 1 ρ (lemma 6.1), w 0 (ξ ) = 4.22e

w ke 1 (ξ ) = O(ξ 2k-3 ), w ke 1 (z) = O(z -2(k-1)
). We end the proof with lemma 3.5:

w ke 1 is an entire function, for every ξ ∈ C, | w ke 1 (ξ )| ≤ 3 √ ρ 2 k 27 4ρ e 27 4ρ |ξ | and for every positive integer 1 ≤ m ≤ 2k -3, | w ke 1 (ξ )| ≤ 3 √ ρ 2 k 27 4ρ m+1 ζ m-1 (m -1)! * e 27 4ρ ζ (|ξ |).
This ends the proof.

Borel-Laplace summability of the transseries

Before keeping on, we lay down a definition, see also [11]. Definition 6.2. Let ( g k ) k≥0 be a sequence of formal series

g k (z) ∈ C[[z -1 ]]. One says that the transseries g(z,V ) = ∞ ∑ k=0 V k g k (z) is Borel-Laplace summable in a direction θ ∈ S 1 if each g k is Borel-Laplace summable in that direction and if the series of functions ∞ ∑ k=0 V k S θ g k (z
) converges uniformaly on any compact subset of a domain of the form

• Π θ τ × V . In that case, one denotes by S θ g(z,V ) ∈ O( • Π θ τ × V
) its sum, called the Borel-Laplace sum of the transseries.

In the sequel, we have in mind to analyze the Borel-Laplace summability of the transseries given by proposition 5.2. This means analyzing the Borel-Laplace summability of the transseries ∑ ∞ k=0 V k w ke 1 (z), resp. ∑ ∞ k=0 V k w ke 2 (z), then substituting V = Ue -z z 3/2 , resp. V = Ue z z 3/2 , in the Borel-Laplace sum. Notice however that the mapping z → e ±z z 3/2 is ill-defined on C but should be considered on the Riemann surface of the square root or on its universal covering C. This justifies the use of domains of the form Π θ τ ∈ C, θ ∈ S 1 (see definition 3.19) in what follows.

Definition 6.3. Let g : C → C and κ : R → R + be two continuous functions, θ ∈ S 1 and τ ∈ R. We set θ (g, τ, κ)

= c>τ {z ∈ Π θ c , |g(z)| < κ(c)} ⊂ C. Let I ⊂ S 1
be an open arc, γ : I → R a locally bounded function and K : I → C 0 (R, R + ) a continuous function. We denote by V (I, g, γ, K ) the domain of C defined as follows:

V (I, g, γ, K ) = θ ∈I θ g, γ(θ ), K (θ ) ⊂ C.
w(z,Ue i ) = ∞ ∑ k=0 [V i (z,U)] k w ke i (z), V i (z,U) = Ue -λ i z z -τ i , i = 1, 2, (6.15)
are Borel-Laplace summable and their Borel-Laplace sums are holomorphic solutions of (3.6). More precisely, for any R > 0, for any open arc I j =] jπ, ( j + 1)π[⊂ S 1 , j ∈ Z, the sum

S I j w(z,Ue i ) := ∞ ∑ k=0 [V i (z,U)] k S • I j w ke i ( • z), (6.16
)

with • I j = π(I j ) ⊂ S 1 and • z = π(z) ∈ C , converges to a function of (z,U) holo- morphic on V (I j ,V i (R), τ, K ) × D(0, R) where one can choose τ(θ ) = 27 4| sin(θ )| and K (θ ) : c ∈ R → 2c 2 3τ(θ ) 2 √ sin(θ )
. Moreover, the sum S I j w is solution of equation (3.6).

Proof. This theorem is a consequence of theorem 3.3, lemma 6.3, lemma 6.4 and lemma 6.5. Let us precise the reasoning for i = 1 and the open arc I 0 =]0, π[⊂ S 1 . We know from lemmas 6.4 and 6.5 (applied with m = 2k -3) that for any δ ∈]0, π 2 [ and any integer k ≥ 2, for every ζ ∈

• ∞ 0 (]δ , π -δ [), | w ke 1 (ζ )| ≤ 3 sin(δ ) 2 k A 2k-2 δ ξ 2k-4 (2k -4)! * e τ δ ξ (ξ ), ξ = |ζ |, (6.17)
with A δ = τ δ = 27 4 sin(δ )

. We now fix a direction θ ∈ I 0 and for k ≥ 2, we consider the Borel-Laplace sum

S • θ w ke 1 ( • z) = ∞e i • θ 0 e - • zζ w ke 1 (ζ ) dζ = +∞ 0 e - Fig. 6.1 The (shaded) domain V (I 0 ,V 1 (0.5), τ, K ) on pro- jection, for τ(θ ) = 27 4| sin(θ )| , K (θ ) (c) = 2c 2 3τ(θ ) 2 √ sin(θ ) and V 1 (z,U) = Ue -z z 3/2 . |S θ w ke 1 (z)| ≤ 3 sin(θ ) 2 k A θ c 2k-2 c c -τ θ .
We turn to the series of function

∑ k≥2 Ue -z z 3/2 k S θ w ke 1 (z). From what pre- cedes, for any R > 0, for any c > c > τ θ , for every (z,U) ∈ Π θ c × D(0, R), the series is normally convergent when |Re -z z 3/2 | ≤ 2c 2 3A 2 θ √ sin(θ )
. We end with theorem 3.3 and lemma 6.3: for any direction θ ∈ I 0 , for any c > τ θ , the series of functions ∑ k≥0 Ue -z z 3/2 k S θ w ke 1 (z) defines a holomorphic function on the do-

main θ × D(0, R) with θ = c>τ θ {z ∈ Π θ c , |Re -z z 3/2 | < 2c 2 3A 2 θ √ sin(θ )
}. Making θ varying on I 0 , these functions glue together to provide a holomorphic function S I 0 w(z,Ue 1 ) on the domain

V (I 0 ,V 1 (R), τ, K ) × D(0, R) with τ(θ ) = 27 4| sin(θ )| and K (θ ) : c ∈ R → 2c 2 3τ(θ ) 2 √ sin(θ )
(since A θ = τ θ ), see Fig. 6.1. Finally, we encourage the reader to show that S I j w solves the ODE (3.6). Remark 6.1. The theorem 6.2 can be shown by other means, see the comments in Sect. 6.5.

Remarks

In what follows we set w tru, j,i (z,U) = S I j w(z,Ue i ).

1. We know by proposition 5.2 that w ke 2 (z) = w ke 1 (-z) for every k ≥ 0. One deduces that for any j ∈ Z, for any θ ∈ I j , for every z ∈ Π π-θ τ(π-θ ) , ze iπ ∈ Π -θ τ(-θ ) and S π-θ w ke 2 (z) = S -θ w ke 1 (ze iπ ). Therefore, for any θ ∈ I j , for every z ∈ Π π-θ τ(π-θ ) , S π-θ w(z,Ue 2 ) = S -θ w(ze iπ ,Ue iπ/2 e 1 ) and, as a consequence, for any j ∈ Z:

for every z ∈ V (I j ,V 2 (U), τ, K ), w tru, j,2 (z,U) = w tru, j-1,1 (ze iπ ,Ue iπ/2 ), for every z ∈ V (I j ,V 1 (U), τ, K ), w tru, j,1 (z,U) = w tru, j-1,2 (ze iπ ,Ue iπ/2 ). (6.18)

2. Here we adopt the convention : for z = re iα ∈ C, we set z = r -iα ∈ C.

We know by proposition 5.2 that w ke i (z) ∈ R[[z -1 ]] for any k ∈ N, i = 1, 2. Thus, for any j ∈ Z and any θ ∈ I j , for z ∈ Π θ τ(θ ) , S θ w ke i (z) = S -θ w ke i (z). Therefore, for any j ∈ Z, for every z ∈ V (I j ,V i (U), τ, K ), w tru, j,i (z,U) = w tru,(-j-1),i (z,U) and with (6.18) we deduce that, for every z ∈ V (I j ,V 1 (U), τ, K ) and z ∈ V (I j ,V 2 (U), τ, K ) respectively,

w tru, j,1 (z,U) = w tru, j,2 ze -(2 j+1)iπ ,Ue -( j+1/2)iπ (6.19)
w tru, j,2 (z,U) = w tru, j,1 ze -(2 j+1)iπ ,Ue -( j+1/2)iπ .

Considerations on the domain

Viewing (6.18) and ( 6. 19), it will be enough for our purpose to consider the domain

V (I 0 ,V 1 , τ, K ) with I 0 =]0, π[, V 1 (U) (z) = Ue -z z 3/2 with |U| > 0, τ(θ ) = 27 4| sin(θ )| , K (θ ) (c) = 2c 2 3τ(θ ) 2 √ sin(θ )
. We would like to describe the boundary of this domain. As a matter of fact, we will restrict ourself to describing its

subdomain θ V 1 (U), τ(θ ), K (θ ) with θ = π/2.
Considered by projection on C, this domain reads: z = x + iy, (x, y) ∈ R 2 , belongs to

π 2 V 1 , τ( π 2 ), K ( π 2 ) if and only if there exists λ > 1 so that      y < - 27 4 λ |U|e -x (x 2 + y 2 ) 3/4 < 2 3 λ 2 .
(We take c = 27 4 λ > τ(π/2)). We now fix y = -27 4 λ with λ > 1 and we remark that z = x + iy belongs to

π 2 V 1 (U), τ( π 2 ), K ( π 2 ) iff x > X with X such that |U|e -X (X 2 + y 2 ) 3/4 = 2 3 4 27 y 2 . (6.20)
Indeed, just see that the real mapping x → e -x (x 2 + y 2 ) p is decreasing when |y| ≥ p, and use an argument of continuity. With the implicit function theorem, these arguments show the existence of a unique solution X : y ∈] -∞, -3 4 [ → X(y) of (6.20), of class C ∞ and increasing with y, which can be described as follows. The above equality is equivalent to writing and we can remark that X(-α -1/2 ) = 0 if -α -1/2 < -3 4 . When assuming y 2 X 2 , we get X = -ln(αy 2 ) 4 +ε, ε = o(1) as a first approximation. Plugging this in (6.21), one gets

X = - ln(αy 2 ) 4 + 3 ln 2 (αy 2 ) 4 2 y 2 + o(y -2 )
and one can keep on this way to get an asymptotic expansion at any order of the solution 3 . To put it in a nutshell: Corollary 6.1. In theorem 6.2, the sum w tru,0,1 (z,U) = S I 0 w(z,Ue 1 ) defines, for any U ∈ C , a holomorphic function with respect to z on a domain which contains, by projection on C, a subdomain of the form z = x + iy, y

< - 27 4 , x > X(y)
where X is an increasing C ∞ function on ] -∞, -3 4 [, whose asymptotics when y → -∞ is given by:

X(y) = - ln(αy 2 ) 4 + 3 ln 2 (αy 2 ) 4 2 y 2 + o(y -2 ), α = 32 2187|U| 4 (6.22)
and so that X(-α -1/2 ) = 0 if -α -1/2 < -3 4 .

Summability of the formal integral

We saw with corollary 5.2 that the formal integral can be interpreted as a formal transformation w = Φ(z, u),

Φ(z, u) = ∑ k∈N 2 u k w [0] k (z), (6.23)
that formally transforms the prepared ODE (3.6) into the normal form equation (5.66). It is then natural to wonder whether this formal transformation gives rise to an analytic transformations Φ θ (z, u) by Borel-Laplace summation,

Φ θ (z, u) = S θ Φ(z, u) = ∑ k∈N 2 u k S θ w [0] k (z),
with a definition of the sum similar to that of definition 6.2. One could give a positive answer to this question, for the price of some further effort.

One has to extend lemma 6.5 to the whole k-th series w

[0] k . It is worth for this matter to complete the Banach spaces detailed by proposition 3.9 by other "focusing algebras" for which we refer to [6], in particular those based on L 1 ν -norms.

This does not mean that the formal integral is Borel-Laplace summable : this is wrong, due to the effect of the exponentials. Only the restrictions of the formal integral to convenient submanifolds is 1-summable, which means here just considering one of the two transseries. However, the sums of the two transseries share no common domain of convergence and a fortiori the formal integral cannot be summed by Borel-Laplace summation. We do not pursue toward this direction and we conclude this chapter with the truncated solutions.

Truncated solutions for the first Painlevé equation

We know from theorem 6.2 that the sum w tru, j,i (z,U) = S I j w(z,Ue i ), j ∈ Z and i = 1, 2, is a holomorphic solution of (3.6), for z on a domain of the form V (I j ,V i (U), τ, K ). From its very definition and from corollary 6.1, the domain V (I j ,V i (U), τ, K ) contains a sectorial neighbourhood of infinity with aperture ( I j,i where (see Fig. 6.1):

• when i = 1, ( I j,1 =] -1 2 π, + 1 2 π[-jπ for j even, ( I j,1 =] -3 2 π, -1 2 π[-jπ for j odd; • when i = 2, ( I j,2 =] -1 2 π, + 1 2 π[-jπ for j odd, ( I j,2 =] -3 2 π, -1 2 π[-jπ for j even.
To go back to the the first Painlevé equation (2.1), we use the transformation T of definition 3.20. Definition 6.4. The conformal mapping T sends the domain V (I, g, γ, K ) onto the domain T V (I, g, γ, K ) and we set

(I, g, γ, K ) = T V (I, g, γ, K ) , • (I, g, γ, K ) = π S (I, g, γ, K ) . (6.24)
The domain (I j ,V i (U), τ, K ) contains a sectorial neighbourhood of infinity with aperture K j,i (see Fig. 6.2):

• when i = 1, K j,1 =] -7 5 π, -3 5 π[- 4
5 jπ for j even, K j,1 =] -11 5 π, -7 5 π[-4 5 jπ for j odd;

• when i = 2, K j,2 =] - 7 5 π, -3 5 π[-4 5 jπ for j odd, K j,2 =] -11 5 π, -7 5 π[-4 5 jπ for j even.

In any case, the domains (I j ,V i (U), τ, K ) are in connection: for every j ∈ Z,

(I j+1 ,V 2 (U), τ, K ) = e -4iπ/5 (I j ,V 1 (U), τ, K ).
From (3.4), (2.6), (2.7), the transformation

z ∈ V (I j ,V i (U), τ, K ) ↔ x ∈ (I j ,V i (U), τ, K ) w tru, j,i (z,U) ↔ u tru, j,i (x,U) = e i π 2 x 1 2 √ 6 1 - 4 25 T -1 (x) 2 + w tri, j,i T -1 (x),U T -1 (x) 2
provides the solutions u tru, j,i (x,U) for the first Painlevé equation. These are the truncated solutions.

The property (6.18) translates into the following relationships between truncated solutions: for any j ∈ Z, for every x

∈ (I j ,V 1 (U), τ, K ), resp. x ∈ (I j ,V 2 (U), τ, K ),
u tru, j,1 (x,U) = e 2iπ/5 u tru, j+1,2 (xe -4iπ/5 ,Ue -iπ/2 ) (6.25) u tru, j,2 (x,U) = e 2iπ/5 u tru, j+1,1 (xe -4iπ/5 ,Ue -iπ/2 )

These are the symmetries discussed in Sect. 2.5. In the same way from (6.19), for any j ∈ Z, for every x ∈ S(

I j ,V 1 (U), τ, K ), respectively x ∈ (I j ,V 2 (U), τ, K ), u tru, j,1 (x,U) = e 2 5 (2 j+1)iπ u tru, j,2 (xe -2 5 (4 j+7)iπ ,Ue -( j+1/2)iπ ), (6.26)
u tru, j,2 (x,U) = e 2 5 (2 j+1)iπ u tru, j,1 (xe -2 5 (4 j+7)iπ ,Ue -( j+1/2)iπ ).

(I 0 ,V 1 (U), τ, K ) for τ(θ ) = 27 4| sin(θ )| , K (θ ) (c) = 2c 2 3τ(θ ) 2 √ sin(θ ) and V 1 (z,U) = Ue -z z 3/2 .

Comments

We mentioned in Sect. 5.5 the existence of formal transforms of the type v = T (z, u),

T (z, u) = ∑ k∈N n u k v k (z), v k (z) ∈ C n [[z -1 ]
] that brings differential and difference systems to their linear normal form, under some convenient hypotheses. For differential equations of type (5.67), the series v k are in general not 1-summable but multisummable [10]. The first results in that direction, concerning the multisummability of the formal series solutions, have been obtained by Braaksma [1] then by Ramis & Sibuya [12]. A resurgent approach for 1-level differential equations is undertaken by Costin in [4], with the proof of the 1-summability of the formal integral on restriction to convenient submanifolds. These results have been generalized to differential and difference equations, see e.g. [2,9,7,5] and references therein, at least for the cases where no resonance occurs. The question of the (multi)summability of the above formal transforms may be delicate, even for 1-level differential systems or ODEs, when quasi-resonance occurs, giving rise to small divisors.

If λ λ λ = (λ 1 , • • • , λ n ) stands
for the multipliers and in absence of resonance, it may happen that λ λ λ .k comes close to one multiplier, for some k ∈ N n . Thus, the construction of the formal integral gives rise to division by small factors. One has "quasi-resonance" when there exists an increasing sequence (k j ∈ N n ) such that lim j→∞ λ λ λ .k j = 0 fast enough, a condition that translates into diophantine relations on the sequence.

More details on this subject can be found in [8]. We finally mention a general upshot, that of the formation of singularities near the anti-Stokes rays. Considering the Borel-Laplace sum of a transseries stemming from (resurgent) 1-level differential or difference equations, it is possible, as shown in [7] (see also [6]) to analyze its behavior on the boundary of its domain of convergence, by a suitable use of a multi-scale analysis. This is detailed in [5] for the first Painlevé equation.

Chapter 7

Supplements to resurgence theory Abstract This chapter is devoted to some general nonsense in resurgence theory which will be useful to study furthermore the first Painlevé equation from the resurgence viewpoint. We define sectorial germs of holomorphic functions (Sect. 7.2) and we introduce the sheaf of microfunctions (Sect. 7.3). This provides an approach to the notion of singularities which is the purpose of Sect. 7.4. We define the formal Laplace transform for microfunctions and for singularities and conversely, the formal Borel transform acting on asymptotic classes (Sect. 7.5). The main properties of the Laplace transform needed in this course are developed to Sect. 7.6. We then introduce some spaces of resurgent functions and define the alien operators (Sect. 7.7 to 7.9).

Introduction

In this introduction, we assume that the reader has a previous acquaintance with 1-summability theory, much discussed in [14] to which we refer.

At its very root, one can rely the Borel-Laplace summation scheme to the simple formula

1 z n = L θ ζ n-1 Γ (n) = ∞e iθ 0 e -zζ ζ n-1 Γ (n) dζ , n ∈ N , z ∈ • Π θ 0 . Let ϕ ∈ O(D(0, R)) be a holomorphic function and ∑ n≥1 a n ζ n-1 Γ (n)
be its Taylor series at the origin. We choose an open arc

I =] -α + θ , θ + α[, 0 < α ≤ π/2,
bisected by the direction θ , and we set

I =] -α -θ , -θ + α[⊆ θ . For some r ≥ 0, we set • ∞ = • ∞ r (I ). For any cut-off κ ∈]0, R[, the truncated Laplace inte- gral ϕ κ (z) = κe iθ 0 e -zζ ϕ(ζ )dζ provides an element of A 1 ( • ∞ ) whose 1-Gevrey asymptotics T 1, • ∞ ϕ κ (z) in • ∞ is given by the 1-Gevrey series ∑ n≥1 a n z n ∈ C[[z -1 ]] 1 .
This is essentially the Borel-Ritt theorem for 1-Gevrey asymptotics. For two cut-off

points κ 1 , κ 2 ∈]0, R[, the difference ϕ κ 1 -ϕ κ 2 belongs to A ≤-1 ( • ∞ ), the differential ideal of A 1 ( • ∞ ) made of 1-exponentially flat functions on • ∞ . One gets this way a morphism L (I) : ϕ ∈ O 0 → cl(ϕ κ ) ∈ A 1 (I )/A ≤-1 (I )
, where here O 0 stands for the constant sheaf (of convolution algebras) over S 1 . By (obvi-129 ous) compatibility with the restriction maps, one obtains 1 a morphism of sheaves of differential algebras, L : O 0 → A 1 /A ≤-1 , where the quotient sheaf A 1 /A ≤-1 over S 1 is known to be isomorphic to the constant sheaf C[[z -1 ]] 1 (Borel-Ritt theorem 3.4, see [14,17]). The formal Laplace transfom L is an isomorphism, the inverse morphism being the formal Borel transform B : C[[z -1 ]] 1 → O 0 (seen as a morphism of sheaves).

One can extend the theory by considering Laplace integrals defined along Hankel contours. For instance, standard formulae provide

Γ (σ ) = 1 1 -e -2iπσ γ [-2π,0],ε e -ζ ζ σ -1 dζ , σ ∈ C \ N, (7.1)
where the integration contour γ [-2π,0],ε is the (endless) Hankel contour drawn on Fig. 7.1, while ζ σ -1 = e (σ -1) log ζ and log ζ is the branch of the logarithm so that arg(log ζ ) ∈] -2π, 0[. Performing a change of variable, one gets the identity

1 z σ = L 0 ∨ I σ (z) = γ [-2π,0],ε e -zζ ∨ I σ (ζ )dζ , z ∈ • Π 0 0 , (7.2)
with z σ = e σ log z where this time log z is the branch of the logarithm so that arg(log z) ∈]π, π[, while

∨ I σ (ζ ) =        ζ σ -1 log(ζ ) 2iπΓ (σ ) for σ -1 ∈ N ζ σ -1 (1 -e -2iπσ )Γ (σ ) for σ -1 ∈ C \ N.
The form of ∨ I σ that we give for σ -1 ∈ C \ N is well-defined when -σ / ∈ N. It can be analytically continued to the case -σ ∈ N by the reflection formula.

This example provides another one that will be used later on : for any m ∈ N, any

σ ∈ C \ N , for z ∈ • Π 0 0 , (-1) m z -σ (log z) m = L 0 ∨ Jσ,m, ∨ Jσ,m= ∂ ∂ σ m ∨
I σ with the above convention for the log z. Remark however that L 0 ∨ I σ = L 0 ∨ I σ +hol when hol is any holomorphic function on a half-strip containing the origin, with at most exponential growth of order 1 at infinity. This justifies the introduction of the spaces of microfunctions and singularities that we do in the next sections.

This chapter can be seen as a sequel of the resurgence theory developed in [18]. For most of the materials presented here, we mainly refer to [7,9,10,1,18], see also [4,24,21]. Another approach to resurgence theory is provided in [27].

Sectorial germs

Sectors

We precise our notations for sectors on ( C, π), the Riemann surface of the logarithm (Compare these notations with definition 3.3).

= {ζ = re iθ | θ ∈ S 1 , 0 < r < R(θ )} ⊂ C.
We simply write 0 for such a domain, when there is no need to specify the function R.

Sectorial germs

(I) = lim -→ R→0 O( • R 0 (I))
the space of germs of direction I at 0, and by O 0 the sheaf over S 1 associated with the presheaf O 0 .

As a rule in this paper for the (pre)sheafs one encounters, the restriction maps are the usual restrictions of functions. We warn the reader that the presheaf O 0 is not a sheaf over S 1 (see for instance a counter example given in [14]) : for an open arc I, a section

∨ ϕ∈ O 0 (I) = Γ (I, O 0 ) is a collection of holomorphic functions ϕ i ∈ O( • R i 0 (I i ))
that glue together on their intersection domains, the set {I i } being an open covering of I Example 7.1. We denote by C{ζ , ζ -1 } the space of Laurent series ∑ n∈Z a n ζ n which converge on a punctured disc D(0, R) . This space can also be seen as a constant sheaf over S 1 and the space O 0 (S 1 ) of global sections of O 0 on S 1 coincides with C{ζ , ζ -1 }.

For n ∈ N and a given direction θ 0 ∈ S 1 , let us consider the sectorial germ

∨ ϕ θ 0 (ζ ) = ζ n-1 log(ζ ) 2iπΓ (n) ∈ O 0 θ 0 ,
for any given determination of the log. Here O 0 θ 0 denotes the stalk at θ 0 of the sheaf O 0 . When making θ varying from θ 0 on

I =] -π + θ 0 , θ 0 + π[⊂ S 1 , the sectorial germs ∨ ϕ θ ∈ O 0
θ glue together and defined a section ∨ ϕ ∈ Γ (I, O 0 ) which cannot be prolonged to a global section.

This last example illustrates the need for defining sectorial germs for functions defined on sectors of C. The covering map π : S 1 → S 1 allows to consider the sheaf π O 0 over S 1 , that is the inverse image by π of the sheaf O 0 (see [1,12,3]). For J an open arc of S 1 , an element ∨ ϕ of π O 0 (J) appears as an element of the space

Γ (J, O 0 ) of multivalued sections of O 0 on J, that is ∨ ϕ = s(J)
where s is a continuous map such that the following diagram commutes:

O 0 = θ ∈S 1 O 0 θ s ↓ S 1 -→ S 1 π
. We say that in another way in the following definition:

Definition 7.3. Let J ∈ S 1 be an open arc. One says that two functions

ϕ 1 ∈ O( R 1 0 (J)), ϕ 2 ∈ O( R 2
0 (J)) define the same sectorial germ ∨ ϕ of direction J at 0 when ϕ 1 and ϕ 2 coincide on a same domain of type 0 (J). We denote by Γ (J, O 0 ) the space of multivalued sections of germs of direction J. Remark 7.1. For any ω ∈ C and by translation, one can of course define O ω , the sheaf over S 1 of sectorial germs at ω, associated with the presheaf O ω .

Microfunctions

We introduce the sheaf of microfunctions C ω at ω ∈ C, in the spirit of [1] to whom we refer. Since C ω is deduced from C = C 0 by translation, we make the focus on the case ω = 0.

Microfunctions, definitions

We complete definition 3.13. Definition 7.4. Let θ be a direction and I =]α, β [ be an open arc (of S 1 or S 1 ). we set:

1. θ * = -θ and I =]β , -α[ the complex conjugate arc; 2.

( θ =] -π 2 -θ , -θ + π 2 [ and ( I = θ ∈I ( θ ; 3. θ =]θ -3π/2, θ -π/2[ the copolar of θ ; 4. Ǐ =]α -3π/2, β -π/2[= θ ∈I θ the copolar of I; 5. when |I| > π, I =]α +π/2, β -π/2[; when |I| < π, I =]β -π/2, α +π/2[. When |I| = π, we set I = {β -π/2}.
We would like to define "microfunctions of codirection I at 0". For any open arc I ⊂ S 1 of length ≤ π, we notice that its copolar Ǐ is of length ≤ π, thus can be seens as an arc of S 1 . For such an arc, we set Ǒ0 (I) = O 0 ( Ǐ). We now remark that for two arcs I 2 ⊆ I 1 of lengths ≤ π, one has Ǐ2 ⊆ Ǐ1 . The restriction map ρ Ǐ2 , Ǐ1 : O 0 ( Ǐ1 ) → O 0 ( Ǐ2 ) gives rise to a restriction map ρI 2 ,I 1 = ρ Ǐ2 , Ǐ1 from Ǒ0 (I 1 ) into Ǒ0 (I 2 ). This justifies the following definition. Viewing O 0 as a constant sheaf over S 1 , we set C = Ǒ0 /O 0 . This quotient sheaf over S 1 is the sheaf of microfunctions at 0 and C (I) = Γ (I, C ) is the space of sections of microfunctions of codirection I at 0. The sheaf of microfunctions C makes allusion to Sato's microlocal analysis, see, e.g. [23,13,19]. We mention that microfunctions depending on parameters can be also defined, see for instance [4] for a resurgent context.

We mention that C (I) = Ǒ0 (I)/O 0 , that is the quotient sheaf coincide with the pre-quotient sheaf, because O 0 is a constant sheaf.

In what follows, we transpose with some abuse the notations for singularities introduced in [18] to that for microfunctions. When I is an arc of length > π, then Ǐ is of length larger than 2π and should be seen as an arc of S 1 . In that case, a microfunction ϕ of C (I) is represented by an element ∨ ϕ of Γ ( Ǐ, O 0 ). For any arc I ⊂ S 1 of length > π, one can define the variation map var :

var : ϕ ∈ C (I) → ϕ ∈ Γ ( I, O 0 ), ϕ(ζ ) = ∨ ϕ (ζ )- ∨ ϕ (ζ e -2iπ ).
∨ I 0 ) equal to ϕ(0)δ (0) = ϕ(0)δ . 2.
More generally, the constant sheaf C{ζ , ζ -1 } over S 1 can be seen as a subsheaf of C (of vector spaces). Any microfunction ψ of C{ζ , ζ -1 } can be written as a sum ∑ n≥0 a n I -n = ∑ n≥0 a n δ (n) , where the Laurent series

∨ ψ (ζ ) = ∑ n≥0 a n (-1) n 2iπ n! ζ n+1
converges for |ζ | > 0. 3. We assume that ϕ ∈ O 0 is a germ of holomorphic function. For any given direction θ 0 ∈ S 1 , we consider the microfunction φ θ 0 = sing θ 0 0 ϕ log 2iπ ∈ C θ 0 (where C θ 0 is the stalk at θ 0 of the sheaf C ), represented by the sectorial germ ∨ φ θ 0 = ϕ log 2iπ ∈ Ǒ0 θ 0 , for any given determination of the log (remark that φ θ 0 does not depend on the chosen determination). Making θ varying from θ 0 up to θ 0 + 2π on S 1 , the microfunctions φ θ = sing θ 0 ϕ log 2iπ ∈ C θ glue together and

φ θ 0 = φ θ 0 +2π
. This provides a global section φ = sing 0 ϕ log 2iπ ∈ Γ (S 1 , C ) which does not depend of the chosen determination of the log one started with. It can be shown (through the variation map) that the space of global sections Γ (S 1 , C ) of the sheaf of microfunctions, is composed of microfunctions of the form ψ +sing 0 ϕ log 2iπ , with ψ ∈ C{ζ , ζ -1 } and ϕ ∈ O 0 , see [1]. 4. We suppose σ -1 ∈ C \ N and let θ ∈ S 1 be a direction. The microfunction

φ θ = sing θ 0 ∨ I σ , represented by the sectorial germ ∨ I σ (ζ ) = ζ σ -1 (1 -e -2iπσ )Γ (σ )
, is well-defined once the determination of the log has been chosen. Let us now fix the arc I =]0, 2π[, consider the arc Ǐ =] -3π/2, 3π/2[ as an arc of S 1 and ∨ I σ ∈ Γ ( Ǐ, O 0 ) as a (uniquely well-defined) multivalued section of O 0 on Ǐ.

One can apply to its associated microfunction I σ ∈ C (I) the variation map and

var( I σ ) = I σ ∈ Γ ( I, O 0 ), I =]π/2, 3π/2[, is given by I σ (ζ ) = ζ σ -1 Γ (σ ) .

Microfunctions and convolution product

This subsection is devoted to convolution products of microfunctions. We start with some geometrical preliminaries.

( I), dist(ζ , • ∞ 0 ( I)) ≤ dist(ξ , η) (dist
is the euclidean distance). Indeed, by the projection theorem for convex sets, there exist a unique point η 0 on the closure of

• ∞ 0 ( I) so that dist(ζ , η 0 ) = dist ζ , • ∞ 0 ( I) , see

ζ-η ζ

h I η 1 J 1 ε 0 δ bisector of the segment [ζ , η 0 ] separates the two convex sets • ∞ 0 ( I) and ζ - • ∞ 0 ( I). Therefore, if dist(ζ , S ε ( I)) ≥ 2ε, then ζ -S ε ( I) ⊂ • S ε (I). Lemma 7.2. Let I =]α, β [⊂ S 1 be an open sector of length ≤ π and ε > 0. We consider η 1 ∈ (-∂ • S ε (I)) and we set r = |η 1 |. We suppose that (ε/r) < 1 and we set δ = arcsin(ε/r) ∈]0, π/2[. 1. if J =]β -π/2, α + π/2 + δ [ is an open sector of length < π, we set h = r sin( J). Then, for any ζ ∈ D(0, h), ζ -η 1 ∈ • ∞ 0 ( Ǐ). 2. if J =]β -π/2, α + π/2 + δ [ is an open sector of length ≤ π/2, then, for any ζ ∈ D(0, r), ζ -η 1 ∈ • ∞ 0 ( Ǐ).
Proof. Left as an easy exercise. Just look at Fig. 7.3.

Convolution product of microfunctions

We pick two microfunctions ϕ and ψ of codirection I, where I is an open arc of length < π. For any strict subarc I 1 I, these microfunctions can be represented by

• S ε (I 1 ) ∩ D(0, R) ⊂ • ¯R+r 0 ( Ǐ1 ). We consider a path Γ = Γ I 1 ,ε,η 1 ,η 2 that follows the oriented boundary -∂ • S ε (I 1 ) from η 1 to η 2 with r < |η 1 | < R, r < |η 2 | < R,
∨ χ I 1 ,ε,η 1 ,η 2 (ζ ) = Γ I 1 ,ε,η 1 ,η 2 ∨ ϕ η ∨ ψ (ζ -η)dη (7.3)
is well-defined for all ζ ∈ • S 2ε (I) ∩ D(0, r) and is holomorphic on this domain (which is non empty since 2ε < r).

Γ I 1 ,ε,η 1 ,η 2 . η 1 η 2 I 1 ζ-η 2 ζ-η 1 I 1 ,ε,η ,η 1 2 Γ ε 0 R r ζ Notice that ∨ χ I 1 ,ε,η 1 ,η 2 can be analytically continued to • S 2ε (I) ∪ D(0, r) when ∨ ψ is holomorphic on D(0, R + r), because |ζ -η| < R + r
for η on the integration contour and ζ ∈ D(0, r). Thus, by linearity, adding to

∨ ψ an element of O D(0, R + r)
results in the addition of an element of O D(0, r) for

∨ χ I 1 ,ε,η 1 ,η 2 . Similarly when ∨ ϕ is holomorphic on D(0, R + r), then ∨ χ I 1 ,ε,η 1 ,η 2 can be analytically continued to • S 2ε (I)∪D(0, r) : through an homotopy in D(0, R), just deform the contour Γ I 1 ,ε,η 1 ,η 2 into an arc Γ running from η 1 to η 2 in {η = se iθ | s ∈]r, R[, θ ∈ Ī} ⊂ S ε ( I); by
Cauchy, the two functions

Γ I 1 ,ε,η 1 ,η 2 ∨ ϕ η ∨ ψ (ζ -η)dη and Γ ∨ ϕ η ∨ ψ (ζ -η)dη coincide for ζ ∈ • S 2ε (I) ∩ D(0,
r), while the second integral is holomorphic on D(0, r).

Replacing η 1 , η 2 by η 1 , η 2 on -∂ • S ε (I 1 ), with r < |η 1 | < R, r < |η 2 | < R, results in modifying ∨ χ I 1 ,ε,η 1 ,η 2 by an element of O D(0, h) for h > 0 small enough: the difference ∨ χ I 1 ,ε,η 1 ,η 2 (ζ )- ∨ χ I 1 ,ε,η 1 ,η 2 = η 1 η 1 + η 2 η 2 ∨ ϕ (η) ∨ ψ (ζ -η)dη (7.4)
can be analytically continued from

• S 2ε (I) ∩ D(0, r) to D(0, h). Indeed, using the condition on ε and by lemma 7.2, we see that for η on the two segment contours

and for ζ ∈ D(0, h) with 0 < h ≤ r sin( I), ζ -η remains in • ∞ 0 ( Ǐ1 ) ∩ D(0, R + r) where ∨ ψ is holomorphic.
∨ χ I 1 ,ε,η 1 ,η 2 and ∨ χ I 1 ,ε ,η 1 ,η 2
. By Cauchy, the difference reads like (7.4) with the same conclusion.

In particular, we can let ε → 0 in the above construction: the family of functions ∨ χ I 1 ,ε,η 1 ,η 2 glue together modulo the elements of O 0 , thus providing a microfunction of codirection I 1 . Making the arcs I 1 ⊂ I recovering I, one sees that these microfunctions glue together to give a microfunction of codirection I. We consider the microfunction ψ 0 = δ ∈ C (S 1 ) that we represent by

∨ ϕ * Γ ∨ ψ (ζ ) = Γ I 1 ,ε,η 1 ,η 2 ∨ ϕ η ∨ ψ (ζ -η)dη (7.5) with Γ = Γ I 1 ,ε,η 1 ,η 2 ,
∨ ψ 0 (ζ ) = ϕ 0 (ζ ) ∨ I 0 (ζ ) = ϕ 0 (ζ ) 2iπζ with ϕ 0 ∈ O D(0, R + r) and subject to the con- dition ϕ 0 (0) = 1. Thus ∨ ϕ * Γ ∨ ψ 0 reads: ∨ ϕ * Γ ∨ ψ 0 (ζ ) = 1 2iπ Γ I 1 ,ε,η 1 ,η 2 ∨ ϕ (η) ϕ 0 (ζ -η) ζ -η dη.
By Cauchy and the residue formula, one easily gets that for all ζ ∈

• ¯R+r 0 ( Ǐ1 ) ∩ D(0, r),

∨ ϕ * Γ ∨ ψ 0 = ∨ ϕ +hol
, where hol can be analytically continued to D(0, r). This implies that ϕ * δ = ϕ . We then consider the integral:

∨ ϕ * Γ ×Γ ∨ ψ (ζ ) = 1 2iπ Γ ×Γ ϕ 0 (ζ -(ξ 1 + ξ 2 ) ζ -(ξ 1 + ξ 2 ) ∨ ϕ (ξ 1 ) ∨ ψ (ξ 2 )dξ 1 dξ 2 , (7.6)
ϕ 0 ∈ O D(0, R + r) , ϕ 0 (0) = 1,
where ). This also allows to use the Fubini theorem:

Γ = Γ I 1 ,ε,η 1 ,η 2 , Γ = Γ I 1 ,ε ,η 1 ,η 2 . We remark that for any (ξ 1 , ξ 2 ) ∈ Γ × Γ one has (ξ 1 + ξ 2 ) ∈ S ε+ε ( I 1 ) ∩ D(0, 2R). Thus
∨ ϕ * Γ ×Γ ∨ ψ (ζ ) = Γ 1 2iπ Γ ϕ 0 (ζ -(ξ 1 + ξ 2 ) ζ -(ξ 1 + ξ 2 ) ∨ ψ (ξ 2 )dξ 2 ∨ ϕ (ξ 1 )dξ 1 = Γ 1 2iπ Γ ϕ 0 (ζ -(ξ 1 + ξ 2 ) ζ -(ξ 1 + ξ 2 ) ∨ ϕ (ξ 1 )dξ 1 ∨ ψ (ξ 2 )dξ 2 .
From the previous considerations, we recognize

∨ ϕ * Γ ×Γ ∨ ψ = ∨ ϕ * Γ ∨ ψ +hol for the first equality, ∨ ϕ * Γ ×Γ ∨ ψ = ∨ ψ * Γ ∨
ϕ +hol for the second equality, where hol is a holomorphic function that can be analytically continued to a neighbourhood of 0. As a consequence,

ϕ * ψ = ψ * ϕ ,
that is the convolution product of microfunctions is commutative. One easily shows in the same way that the convolution product of microfunctions is associative. The fact that ∂ is a derivation is obvious.

We have previously seen two kind of integral representations,

1 0 ϕ(η) ∨ ψ (ζ -η)dη and η 2 0 ϕ(η) ∨ ψ (ζ -η)dη (7.7)
with η 1 , η 2 as for definition 7.8.

The proof is left as an exercise. (See [24]). Starting with the integral representation (7.5), the idea is to decompose the path Γ I 1 ,ε,η 1 ,η 2 as on Fig. 7.5 and to use the integrability of the log at the origin.

Space of singularities

The reader will recognize in what follows classical notions and notations in resurgence theory already encountered in [18], see also [9,10,24,21].

Γ I 1 ,ε,η 1 ,η 2 . η 1 η 2 I 1 ζ-η 2 ζ-η 1 I 1 ,ε,η ,η 1 2 Γ r R 0 ζ 7.4.

Singularities

Definition 7.9. Let θ ∈ R be a direction and α > 0. We denote by ANA θ ,α the space of sections Γ ( J, O 0 ) where J =]θα -2π, θ + α[⊂ S 1 , and by ANA = Γ ( S 1 , O 0 ) the space of global sections. Thus, ANA is the space of sectorial germs at 0 that are represented by functions ∨ ϕ holomorphic on a simply connected domain of the form 0 . Definition 7.10. One sets SING θ ,α = ANA θ ,α /O 0 and SING = ANA/O 0 . The elements of these quotient spaces are called singularities at 0. One denotes by sing 0 the canonical projection,

sing 0 : ANA → SING ∨ ϕ → ϕ , sing 0 : ANA θ ,α → SING θ ,α ∨ ϕ → ϕ . If sing 0 ( ∨ ϕ ) = ϕ , then ∨ ϕ is called a major of the singularity ϕ .
In particular, with these notations:

Proposition 7.3. The space of singularities SING θ ,α can be identified with the space Γ (J, C ) of multivalued sections of C by π, Notice that SING θ ,α and SING are naturally O 0 -modules.

with J =] -π 2 -α + θ , θ + α + π 2 [.
Definition 7.12. Let f ∈ O 0 be a germ of holomorphic functions and let ϕ = sing 0 ∨ ϕ be a singularity in SING, resp. SING θ ,α . One defines the product f ϕ in SING, resp.

SING θ ,α , by f ϕ = sing 0 ( f ∨ ϕ ).
Definition 7.13. The so-called variation map is defined by: var :

SING → ANA ϕ = sing 0 ( ∨ ϕ ) → ϕ, ϕ(ζ ) = ∨ ϕ (ζ )- ∨ ϕ (ζ e -2iπ ),
and ϕ = var( ϕ ) is called the minor of the singularity ϕ .

The variation map var operates similarly on every element ϕ ∈ SING θ ,α , with

ϕ = var( ϕ ) in Γ ( J, O 0 ), where J =]θ -α, θ + α[⊂ S 1 .
A minor is said to be regular when it belongs to O 0 .

We illustrate the notion of singularities by the following examples. (The reader will recognize sectorial germs used in the introduction of this chapter).

Definition 7.14. The singularities I σ , Jσ,m∈ SING, σ ∈ C, m ∈ N are defined as follows.

• For σ ∈ C \ N , I σ = sing 0 ( ∨ I σ ) where ∨ I σ (ζ ) = ζ σ -1 (1-e -2iπσ )Γ (σ ) .
In particular,

I -n = δ (n) = sing 0 (-1) n 2iπ n! ζ n+1 , n ∈ N. • For n ∈ N , I n = sing 0 ( ∨ I n ) with ∨ I n (ζ ) = ζ n-1 log(ζ ) 2iπΓ (n) . • For m ∈ N and σ ∈ C, Jσ,m= ∂ ∂ σ m I σ .
It is useful to define the following subspaces of "integrable singularities", SING int ⊂ SING and SING int θ ,α ⊂ SING θ ,α . Definition 7.15. An integrable minor is a germ ϕ ∈ ANA holomorphic in the domain 0 ⊂ C which has a primitive φ such that φ → 0 uniformaly in any proper subsector ¯ 0 0 . The space of integrable minors is denoted by ANA int . An integrable singularity is a singularity ϕ ∈ SING which admits a major

∨ ϕ holo- morphic in the domain 0 ⊂ C such that lim ζ →0 ζ ∨ ϕ (ζ ) = 0

uniformaly in any proper

subsector ¯ 0 0 . One denotes by SING int the space of integrable singularities.

There is a natural injection O 0 → ANA int from the space of germs of holomorphic functions to the space ANA int of integrable minors. The space ANA int can be equipped with a convolution product, by extending the usual law convolution on O 0 .

It is not hard to show that integrable singularities satisfy the following property:

Proposition 7.4. By restriction, the variation map var induces a linear isomorphism SING int → ANA int . The inverse map is denoted by : ϕ ∈ ANA int → ϕ ∈ SING int .

This allows to transports the convolution law from ANA int to SING int by the variation map.

Definition 7.16. The convolution product of ϕ 1 , ϕ 2 ∈ ANA int is defined by

ϕ 1 * ϕ 2 (ζ ) = ζ 0 ϕ 1 (η) ϕ 1 (ζ -η)dη. The convolution of two integrable singular- ities ϕ 1 = ϕ 1 , ϕ 2 = ϕ 2 ∈ SING int is given by : ϕ 1 * ϕ 2 = ϕ 1 * ϕ 2 .
Quite similarly: Definition 7.17. A minor ϕ holomorphic on the domain 0 ( I) ⊂ C is said to be integrable if ϕ has a primitive φ such that φ → 0 uniformaly in any proper subsector ¯ 0 0 ( I). One denotes by ANA int θ ,α the space of these integrable minors. An integrable singularity is a singularity ϕ ∈ SING θ ,α which has a major proper subsector ¯ 0 0 ( Ǐ). One denotes SING int θ ,α the space of these integrable singularities.

Proposition 7.5. By restriction, the variation map var induces a linear isomorphism SING int θ ,α → ANA int θ ,α . The inverse map is denoted by : ϕ ∈ ANA int θ ,α → ϕ ∈ SING int θ ,α . We end with further definitions. Definition 7.18. Any singularity ϕ of the form ϕ = aδ + ϕ with ϕ ∈ O 0 is said to be simple. The space of simple singularities is denoted by SING simp . The space SING s.ram of simply ramified singularities is the vector space spanned by SING simp and the set of singularities { I -n , n ∈ N}.

Convolution product of singularities

The resurgence theory asserts that the space of singularities SING can be equipped with a convolution product [7,8,18], see also [1,22]. Since SING θ ,α can be identified with the space Γ (J, C ) of multivalued sections of C by π, with J =] -π 2α + θ , θ + α + π 2 [, the convolution product for microfunctions (proposition 7.1) allows to transport this product to SING θ ,α : for any two singularities ϕ , ψ ∈ SING θ ,α and any strict subarc I J of length < π, one can find two ma-

∨ ϕ * Γ ∨ ψ (ζ ) = Γ ∨ ϕ η ∨ ψ (ζ -η)dη (7.8)
or by

∨ ϕ * Γ ×Γ ∨ ψ (ζ ) = 1 2iπ Γ ×Γ f (ζ -(ξ 1 + ξ 2 )) ζ -(ξ 1 + ξ 2 ) ∨ ϕ (ξ 1 ) ∨ ψ (ξ 2 )dξ 1 dξ 2 , (7.9)
with f ∈ O 0 and f (0) = 1 (cf. (7.5) and (7.6)), where Γ = Γ I,ε,η 1 ,η 2 is as in definition 7.7. When considering a covering of J by such arcs I, these sections glue together to give the convolution product ϕ * ψ as a multivalued section of C over J.

Proposition 7.6. The space SING can be equipped with a convolution product denoted by * that makes it a commutative convolution algebra, with unit δ = sing 0 1 2iπζ = I 0 . Moreover:

1. the linear operator, ∂ :

ϕ = sing 0 ( ∨ ϕ ) ∈ SING → ∂ ϕ = sing 0 (-ζ ∨ ϕ ) ∈ SING, is a derivation.
2. if ϕ and ψ belong to SING int , then ϕ * ψ belongs to SING int and ϕ * ϕ = ( ϕ * ϕ).

In particular, the space of simple singularities SING simp is a convolution subalgebra.

Theses properties remain true when one considers SING θ ,α instead of SING.

Proof. We have already shown that SING θ ,α (thus SING) is a commutative convolution algebra for the convolution product with unit δ . The equality ϕ * ϕ = ( ϕ * ϕ) for integrable singularities, emerges from considerations on integrals and is left as an exercise. (Start with proposition 7.2. See [24]).

7.5 Formal Laplace transform, formal Borel transform

Formal Laplace transform for microfunctions

We start with the following definition. Remark 7.2. The fact that A ≤0 is indeed a sheaf of differential algebras is an exercise left to the reader. (We stress that the derivation considered is the usual one for holomorphic functions). The sheaf A ≤0 should not be confused with the sheaf A <0 of flat germs at infinity (definition 3.17). As a matter of fact, A <0 (I) ⊂ A (I) ⊂ A ≤0 (I) where A stands for the presheaf of asymptotic functions (see definition 3.17 and [14,16,17]). We mention that our definition of A ≤0 differs from that of Malgrange in [16] where A ≤0 is defined as the sheaf of sectorial germs that admit an asymptotics belonging to the formal Nilsson class, that is of the form ∑ w(z) log m (z)

z σ , σ ∈ C, m ∈ N, w ∈ C[[z -1 ]].
Our sheaf A ≤0 contains this sheaf as a subsheaf. However, the constructions in the sequel resemble in much aspects to that of Malgrange [16].

The following Lemma is left to the reader as an exercise. This will allow us in a moment to properly define the quotient sheaf A ≤0 /A ≤-1 over S 1 . Definition 7.20. Let θ be any direction (of S 1 or S 1 ). We denote by R θ the ray ]0, e iθ ∞[. For κ > ε ≥ 0, we set R θ ,ε =]εe iθ , e iθ ∞[ and R θ ,ε;κ =]εe iθ , κe iθ [. For any closed arc J = [θ 1 , θ 2 ], we denote by γ J,ε , resp. γ J,ε;κ , the Hankel contour, resp. truncated Hankel contour, which consists in following: We consider another open arc I 2 =]α 2 , β 2 [, Ī2 ⊂ I 1 , so that Ǐ1 \ Ǐ2 has two connected components. We choose one arbitrary direction in each component,

1. R θ 1 ,ε , resp. R θ 1 ,ε;κ ,
θ 1 ∈]α 1 -3π/2, α 2 -3π/2[, θ 2 ∈]β 2 -π/2, β 1 -π/2[. For R > κ > ε > 0, we con- sider the truncated Laplace integral ϕ θ 1 ,θ 2 ,κ (z) = γ [θ 1 ,θ 2 ],ε;κ e -zζ ∨ ϕ (ζ )dζ , see Fig. 7.6.
The function ϕ θ 1 ,θ 2 ,κ satisfies the following properties:

• ϕ θ 1 ,θ 2 ,κ is an entire function, since one integrates on a (relatively) compact path of the domain of holomorphy of ∨ ϕ . -for all z ∈ C,

δ [θ 1 ,θ 2 ],ε e -zζ ∨ ϕ (ζ )dζ ≤ ε| Ǐ1 |Me ε|z| where | Ǐ1 | = β 1 -α 1 + π;
-for any r > 0, for every z ∈

• Π θ 1 r , R θ 1 ,ε;κ e -zζ ∨ ϕ (ζ )dζ ≤ κMe -εr . Similarly, for every z ∈ • Π θ 2 r , R θ 2 ,ε;κ e -zζ ∨ ϕ (ζ )dζ ≤ κMe -εr .
-the domain From this analysis, since ε > 0 can be chosen arbitrarily small, we retain that ϕ θ 1 ,θ 2 ,κ belongs to the space A ≤0 ( • ∞ r (I 2 )), r > 0 large enough. • Furthermore, looking at the above analysis and by Cauchy, we may observe that for two cut-off points κ, κ ∈]ε, R[, for two directions

θ 1 ∈]α 1 -3π/2, α 2 -3π/2[, θ 2 ∈]β 2 -π/2, β 1 -π/2[ the difference ϕ θ 1 ,θ 2 ,κ -ϕ θ 1 ,θ 2 ,κ belongs to A ≤-1 ( • ∞ r (I 2 ))
with r > 0 large enough. We finally remark that adding to ∨ ϕ a function holomorphic on D(0, R) only affects ϕ θ 1 ,θ 2 ,κ (z) by the addition of an element of

A ≤-1 ( • ∞ r (I 2 )), r > 0 large enough.
In this way, one obtains a morphism, L (I,

I 2 ) : ϕ ∈ C (I) → ϕ ∈ A ≤0 (I 2 )/A ≤-1 (I 2 ), ϕ = cl(ϕ θ 1 ,θ 2 ,κ
), which is obviosuly compatible with the restriction maps. This allows to move up to stalks, L α : C α → A ≤0 /A ≤-1 α * and finally 2 to a morphism of sheaves L : C → A ≤0 /A ≤-1 . Definition 7.21. One calls formal Laplace transform for microfunctions at 0, the morphism of sheaves L : C → A ≤0 /A ≤-1 . The quotient sheaf A ≤0 /A ≤-1 over S 1 is called the sheaf of asymptotic classes. An asymptotic class is usually denoted by ϕ .

The term "sheaf of asymptotic classes" is borrowed from [1] where the sheaf A ≤0 is denoted by E 0 , and the sheaf A ≤-1 is denoted by E -. The notation ϕ is own. • since ϕ belongs to A ≤0 ( • ∞ r (I 1 )), we know that for any proper subsector

• ¯∞ r 1 (J )
• ∞ r (I 1 ) and any ε > 0, there exists C > 0 so that, for all z ∈

• ¯∞ r 1 (J ), |ϕ(z)| ≤ Ce ε|z| . Therefore ∨ ϕ z 1 ,α belongs to O( • Π α+π ε ) when z 1 ∈ • ¯∞ r 1 (J )
and α ∈ J . Making α varying in J and since ε > 0 can be chosen arbitrarily small, these functions glue together by Cauchy, and provide a holomorphic function • when ϕ belongs to A ≤-1 (I ), one easily sees from the above analysis that ∨ ϕ z 1 ,I is holomorphic on a domain containing a full neighbourhood of the origin, thus by localisation, an element of O 0 .

∨ ϕ z 1 ,J on • D(J , 0) = • ∞ 0 ( J). Notice that for two points z 1 , z 2 ∈ • ¯∞ r 1 (J ), the differ- ence ∨ ϕ z 2 ,J - ∨ ϕ z 1 ,
To conclude, we have defined a morphism (of C-differential algebras),

B(I ) : ϕ ∈ A ≤0 (I )/A ≤-1 (I ) → ϕ = cl ( ∨ ϕ z 1 ,I ) ∈ C (I)
O 0 → C L ↓↑ B B ↑↓ L A 1 /A ≤-1 →A ≤0 /A ≤-1 .

Formal Laplace transform for singularities and back to convolution product

In the sequel, we translate to singularities what have obtained so far for microfunctions.

Formal Laplace transform for singularities

We start with two definitions.

Definition 7.23. Let θ ∈ S 1 be a direction and α > 0. We denote by ASYMP θ ,α the space of asymptotic classes defined as multivalued sections of A ≤0 /A ≤-1 on J =]π/2αθ , -θ + α + π/2[. We denote by ASYMP the space of asymptotic classes given by global sections of A ≤0 /A ≤-1 on S 1 .

Definition 7.24. Let σ ∈ C be a complex number and m ∈ N. We denote by I σ ∈ ASYMP the asymptotic class represented by 1/z σ . We denote by Jσ,m∈ ASYMP the asymptotic class represented by (-1) m log m (z) z σ . We often simply write 1/z σ instead of I σ and similarly for Jσ,m.

We have already said that the space of singularities SING θ ,α can be identified with the space Γ (J, C ) of multivalued sections of C by π, with J =] -π 2α + θ , θ + α + π 2 [. The formal Laplace transform for microfunctions thus extends to singularities, by inverse image:

L β ∈S 1 C β → β ∈S 1 A ≤0 /A ≤-1 β s ↓ ↓ S 1 ⊃ J β -→ S 1 • β → S 1 • β π
When returning to the very construction of the formal Laplace transform (Sect. 7.5.1), one sees that for any singularity ϕ ∈ SING θ ,α , for any direction

β ∈ J =] -α + θ , θ + α[, setting ( β =] -π 2 + β , β + π 2 [, the formal Laplace trans- form L ( ( β ) ϕ is given as the class ϕ = cl(ϕ β -2π,β ,κ ) ∈ A ≤0 ( ( β )/A ≤-1 ( ( β ) where ( β =] -π 2 -β , -β + π 2 [ and ϕ β -2π,β ,κ (z) = γ [β -2π,β ],ε;κ e -zζ ∨ ϕ (
β ) : SING θ ,α → A ≤0 ( ( β )/A ≤-1 ( ( β ) is called the formal Laplace transform in the direction β ∈ J =] -α + θ , θ + α[. (
For any singularity ϕ ∈ SING θ ,α , one denotes by L J ϕ ∈ ASYMP θ ,α the asymptotic class given by the collection L β ϕ β ∈ J .

Example 7.4. We continue the example 7.3 but for the fact that we now consider Jσ,m as a singularity in SING 0,π . The formal Laplace transform L ]-π,π[ Jσ,m is the asymptotic class Jσ,m∈ ASYMP 0,π seen by restriction as an element of Γ (] -3π/2, 3π/2[, A ≤0 /A ≤-1 ).

We linger for a moment at the cases of singularities of the form ϕ = ϕ ∈ SING int θ ,α . For any direction β ∈]α + θ , θ + α[, the formal Laplace transform ϕ = L β ϕ ,

ϕ ∈ A ≤0 ( ( β )/A ≤-1 ( ( β )
, can be represented by the function

ϕ β -2π,β ,κ (z) = γ [β -2π,β ],ε;κ e -zζ ∨ ϕ (ζ )dζ = R β ,0;κ e -zζ ϕ(ζ )dζ , (7.10)
and we thus recover the "usual" formal Laplace transform (see Sect. 7.1). In particular, we recall that we have extended the convolution law to SING int θ ,α by the variation map: for

ϕ 1 = ϕ 1 , ϕ 2 = ϕ 2 ∈ SING int θ ,α , ϕ 1 * ϕ 2 = ϕ 1 * ϕ 2 .
The above remark (7.10) shows that L β ( ϕ 1 * ϕ 2 ) = (L β ϕ 1 )(L β ϕ 2 ), by the properties of the "usual" formal Laplace transform.

We now assume that ϕ is a simple singularity, ϕ = aδ + ϕ ∈ SING simp with ϕ ∈ O 0 . For any open arc J =]α + θ , θ + α[, the formal Laplace transform ϕ = L J (aδ + ϕ ) is an asymptotic class which belongs to Γ (J , A 1 /A ≤-1 ). This again comes from (an analogue of) the identity (7.10) and classical arguments recalled in the introduction of this chapter. Definition 7.26. One denotes by ASYMP simp the subspace of asymptotic classes obtained by injection of the global sections Γ ( S 1 , A 1 /A ≤-1 ) into ASYMP.

Proposition 7.9. The restriction of the formal Laplace transform L to SING simp has ASYMP simp for its range. Remark 7.4. Consider a formal series ϕ ∈ C[[z -1 ]] and an open arc of the form J =]π/2αθ , -θ + α + π/2[⊂ S 1 . By the Borel-Ritt theorem, there are infinitely many ϕ ∈ A (J ) whose Poincaré asymptotics T (J )ϕ is given by ϕ on J . These various ϕ differ by flat germs, that is elements of A <0 (J ). Therefore as a rule, these germs ϕ represent different asymptotic classes ϕ ∈ ASYMP θ ,α . Now suppose that ϕ is 1-Gevrey and choose a (good) covering (I i ) of J where each I i is an open arc of length less than π. By the Borel-Ritt theorem for 1-Gevrey asymptotics and for each subscript i, there exists ϕ i ∈ A 1 (I i ) whose 1-Gevrey asymptotics T 1 (I i )ϕ i is ϕ. Moreover, each ϕ i is uniquely defined this way up to 1-exponentially flat germs, that is up to elements of A ≤-1 (I i ). One thus gets a uniquely defined section ϕ ∈ Γ (J , A 1 /A ≤-1 ) that can be thought of as an asymptotic class. One can characterize another way this asymptotic class ϕ ∈ ASYMP simp by settling ϕ = L (aδ + ϕ ) where ϕ = ϕ with ϕ the minor of ϕ while a is its constant term.

Definition 7.27. The mapping :

ϕ ∈ C[[z -1 ]] 1 → ϕ = ϕ ∈ ASYMP simp
is defined by ϕ = L (aδ + ϕ ) where ϕ = ϕ, whereas ϕ stands for the minor of ϕ and a its constant term.

Obviously, the mapping is an isomorphism, the inverse map being the 1-Gevrey Taylor map. This allows to merge ϕ with ϕ in practice.

Back to convolution product

We have said without proof that L and B are morphisms of sheaves of algebras. It is thus certainly worthy to prove the following proposition. Proposition 7.10. For any two singularities ϕ 1 , ϕ 2 ∈ SING θ ,α and any direction β ∈]α + θ , θ + α[, the following properties hold:

(L β ϕ 1 )(L β ϕ 1 ) = L β ( ϕ 1 * ϕ 2 ) and L β ( ∂ ϕ 1 ) = ∂ L β ϕ 1 .
Proof. (Adapted from [1]). Let ϕ 1 , ϕ 2 ∈ SING θ ,α be two singularities with majors ∨ ϕ 1 , ∨ ϕ 2 . We pick a direction β ∈ J =]α + θ , θ + α[ and we consider the formal Laplace transforms ϕ 1 = L β ϕ 1 and ϕ 2 = L β ϕ 2 . These are elements of A ≤0 ( ( β )/A ≤-1 ( ( β ) which can be represented respectively by

ϕ 1 (z) = γ 1 e -zζ ∨ ϕ 1 (ζ )dζ ∈ A ≤0 ( • ∞ r ( ( β )), ϕ 2 (z) = γ 2 e -zζ ∨ ϕ 2 (ζ )dζ ∈ A ≤0 ( • ∞ r ( ( β )), with γ 1 = γ [β -2π,β ],ε 1 ;κ 1 , γ 2 = γ [β -2π
,β ],ε 2 ;κ 2 and some r > 0 large enough. The prod-

uct ϕ 1 ϕ 2 ∈ A ≤0 ( ( β )/A ≤-1 ( ( β ) is thus represented by ϕ 1 ϕ 2 (z) = γ 1 ×γ 2 e -z(ζ 1 +ζ 2 ) ∨ ϕ 1 (ζ 1 ) ∨ ϕ 2 (ζ 2 )dζ 1 dζ 2 ∈ A ≤0 ( • ∞ r ( ( β )).
Let us look at the formal Borel transform B(

( β )( ϕ 1 ϕ 2 ) ∈ C ( ( β )
. This Borel transform can be represented by the integral with ε > ε 1 + ε 2 , we can apply Fubini.

∨ (ϕ 1 ϕ 2 ) z 1 ,α 1 (ζ ) = - 1 2iπ R α 1 ,z 1 e zζ ϕ 1 ϕ 2 (z)dz with z 1 ∈ • ∞ r 1 ( ( β ),
Remark that ζ 1 + ζ 2 (or rather

• ζ 1 + • ζ 2 ) remains in the bounded strip {ζ ∈ C | dist(ζ , e iβ [0, κ]) ≤ ε 1 + ε 2 }, for (ζ 1 , ζ 2 ) ∈ γ 1 × γ 2 . Thus ζ -(ζ 1 + ζ 2 ) remains in the domain • Π α 1 +π ε for ζ ∈ • Π α 1 +π 2ε
and this ensures the integrability conditions. This way, we get:

∨ (ϕ 1 ϕ 2 ) z 1 ,α 1 (ζ ) = - 1 2iπ R α 1 ,z 1 e zζ γ 1 ×γ 2 e -z(ζ 1 +ζ 2 ) ∨ ϕ 1 (ζ 1 ) ∨ ϕ 2 (ζ 2 )dζ 1 dζ 2 dz = γ 1 ×γ 2 e z 1 (ζ -ζ 1 -ζ 2 ) 2iπ(ζ -ζ 1 -ζ 2 ) ∨ ϕ 1 (ζ 1 ) ∨ ϕ 2 (ζ 2 )dζ 1 dζ 2 = γ 1 γ 2 e z 1 (ζ -ζ 1 -ζ 2 ) 2iπ(ζ -ζ 1 -ζ 2 ) ∨ ϕ 2 (ζ 2 )dζ 2 ∨ ϕ 1 (ζ 1 )dζ 1
Returning to the very construction of the convolution product for singularities, we see that Example 7.5. We know by theorem 3.3 that the formal series w (0,0) solution of the prepared ODE (3.6) associated with the first Painlevé equation, is 1-Gevrey. Its minor w (0,0) = B w (0,0) is thus a germ of holomorphic functions at the origin and we set w (0,0) = w (0,0) ∈ SING simp . We now consider the singularity I σ * w (0,0) ∈ SING, for any σ ∈ C. By proposition 7.10, for an arbitrary direction β ∈ S 1 , the formal Laplace transform

∨ (ϕ 1 ϕ 2 ) z 1 ,α 1 is
L β I σ * w (0,0) ∈ A ≤0 ( ( β )/A ≤-1 ( ( β )
is the asymptotic class of direction ( β which reads also as:

L β I σ * w (0,0) = L β I σ L β w (0,0) .
On the one hand, L β I σ is the asymptotic class I σ ∈ Γ ( ( β , A ≤0 /A ≤-1 ). On the other hand, L β w (0,0) = w (0,0) . Therefore, L β I σ * w (0,0) = I σ w (0,0) that can be identified with 1 z σ w (0,0) with the branch of z σ determined by the condition arg z ∈ ( β .

Example 7.6. We now use the notations of Sect. 3.4.2.2 but for the fact that we consider arcs on S 1 . We write I 0 =]0, π[ and I 0 =]-3π/2, π/2[⊂ S 1 and in what follows with think of the Laplace-Borel sum w tri,0 = S I 0 w (0,0) as (representing) a multivalued section of A 1 on I 0 . Similarly, we set I 1 =]π, 2π[ and

I 1 =] -5π/2, -π/2[⊂ S 1
and think of w tri,1 = S I 1 w (0,0) as an element of Γ (I 1 , A 1 ). Notice that

I 0 ∩ I 1 =] -3π/2, -π/2[ on S 1 .
Since both w tri,0 and w tri,1 are asymptotic to the 1-Gevrey series w (0,0) , we know that the difference w tri,0w tri,1 is a multivalued section of A ≤-1 on I 0 ∩ I 1 . Therefore, for any σ ∈ C, 1 z σ w tri,0 and 1 z σ w tri,1 glue together to give a multivalued section of A ≤0 /A ≤-1 on I 0 ∪ I 1 , that can be identified with the asymptotic class I σ w (0,0) ∈ ASYMP π,π . The formal Borel transform B(I 0 ) I σ w (0,0) is the multivalued section of C on I 0 =]π/2, 3π/2[ which can be thought of as a singularity in SING π/2,π/2 , and is given by B(I 0 ) I σ w (0,0) = I σ * w (0,0) . Similarly, the formal Borel transform B(I 1 ) I σ w (0,0) is the multivalued section of C on I 1 =]π/2, 5π/2[ which provides a singularity in SING 3π/2,π/2 , of the form B(I 1 ) I σ w (0,0) = I σ * w (0,0) . These two singularities glue together as the element I σ * w (0,0) of SING π,π .

Formal Laplace transform for singularities at ω

The spaces SING ω , resp. SING ω,θ ,α of singularities at ω ∈ C are the translated of SING, resp. SING θ ,α . (See definition 7.11). By its very construction, the formal Laplace transform brings the translation into the multilplication by an exponential.

Definition 7.28. The formal Laplace transform L sends SING ω , resp. SING ω,θ ,α , onto the space denoted by e -ωz ASYMP, resp. e -ωz ASYMP θ ,α , made of asymptotic classes with support based at ω.

We mention the following result that can be thought of as an analogue of the Watson's lemma [14].

Lemma 7.4. For any ω ∈ C , the sum of the two C-vector spaces ASYMP θ ,α and e -ωz ASYMP θ ,α is direct.

Proof. We consider an asymptotic class ϕ ∈ ASYMP θ ,α . By definition, one can find a (good) open covering (J j ) of J =]π/2αθ , -θ + α + π/2[ and a "0cochain" ϕ j ∈ A ≤0 (J j ) j with associated "1-coboundary" ϕ j+1ϕ j ∈ A ≤-1 (J j+1 ∩ J j ) j that represents ϕ . Now assume that ϕ also belongs to e -ωz ASYMP θ ,α . Considering a refinement of (J j ) if necessary, one deduces that ϕ j ∈ A ≤-1 (J j ) for at least one j, since J is an arc of length > π. This implies that the formal Borel transform ϕ ∈ SING θ ,α has a major ∨ ϕ which can be analytically continued to 0, thus ϕ = 0 and as a consequence ϕ = 0.

Laplace transforms

We develop here only matters convenient for this course. For more general nonsense on Laplace transforms in the framework of resurgent analysis, see [1,2,7,8,16].

ϕ I 1 (z) = γ [θ 1 ,θ 2 ],ε e -zζ ∨ ϕ (ζ )dζ = - R θ 1 ,ε + δ [θ 1 ,θ 2 ],ε + R θ 2 ,ε ∨ ϕ (ζ )dζ ,
where Ǐ1 =]θ 1 , θ 2 [ (for the contour of integration, see definition 7.20). This Laplace integral can be decomposed as follows:

• by classical arguments, the integral τ and we observe that for any r > τ, for every z ∈

• Π θ 1 r , R θ 1 ,ε e -zζ ∨ ϕ (ζ )dζ ≤ ∞ ε e -sr Ce τs ds ≤ C r -τ e -ε(r-τ) .
In the same way, the integral

R θ 2 ,ε e -zζ ∨ ϕ (ζ )dζ defines a holomorphic function on • Π θ 2
τ and for any r > τ, for every z

∈ • Π θ 2 r , R θ 2 ,ε e -zζ ∨ ϕ (ζ )dζ ≤ C r -τ e -ε(r-τ) ;
• the integral

δ [θ 1 ,θ 2 ],ε
e -zζ ∨ ϕ (ζ )dζ defines an entire function and

δ [θ 1 ,θ 2 ],ε e -zζ ∨ ϕ (ζ )dζ ≤ C| Ǐ1 |εe τε e ε|z| .
• by arguments already encounter (see Sect. We return to the construction we did to get the formal Borel transform, Sect.

∨ ϕ z 1 ,α (ζ ) = - 1 2iπ R α,z 1
e zζ ϕ(z)dz.

We have seen that, making α varying, one gets an element of Ě ≤1 (I), while ∨ ϕ z 1 ,α depends on z 1 only modulo an element of O(C) ≤1 . We thus get a morphism of sheaves B : A ≤0 → Ě ≤1 /O(C) ≤1 which has the following property (we refer to [1] for the proof): We recall that SING θ ,α can be identified with the space Γ (J, C ) of multivalued sections of C over J =]π/2α + θ , θ + α + π/2[⊂ S 1 . In particular, any singularity ϕ ∈ SING θ ,α can be represented by a major

∨ ϕ ∈ ANA θ ,α = Γ ( J, O 0 ), with J =]θ -α -2π, θ + α[⊂ S 1 . Definition 7.31. An element ∨ ϕ ∈ ANA θ ,α = Γ ( J, O 0 ) is said summable in the direc- tion β ∈ J =] -α + θ , θ + α[ if there exists a neighbourhood J 1 ⊂ J of β so that the two restrictions ∨ ϕ 1 ∈ Γ ( J 1 , O 0 ) and ∨ ϕ 2 ∈ Γ ( J 2 , O 0 ) of ∨
ϕ over J 1 and J 2 = -2π + J 1 respectively, can be represented by elements of Γ ( J 1 , E ≤1 ) and Γ ( J 2 , E ≤1 ) respectively. A singularity ϕ ∈ SING θ ,α is summable in the direction J if for any β ∈ J, the singularity ϕ has a major ∨ ϕ ∈ ANA θ ,α which summable in the direction β . We denote by SING sum θ ,α the space of singularities ϕ ∈ SING θ ,α which are summable in the direction J.

Laplace transforms of summable singularities

We consider a singularity ϕ ∈ SING sum θ ,α and a direction β ∈ J =]α + θ , θ + α[. Let ∨ ϕ be a major of ϕ which is summable in the direction β and set ϕ = var ϕ . Using the notations of definition 7.31, we consider the following Laplace integral where ε > 0 is chosen small enough:

ϕ β (z) = γ [β -2π,β ],ε e -zζ ∨ ϕ (ζ )dζ (7.11) = δ [β -2π,β ],ε e -zζ ∨ ϕ (ζ )dζ - R β -2π,ε e -zζ ∨ ϕ 2 (ζ )dζ + R β ,ε e -zζ ∨ ϕ 1 (ζ )dζ = δ [β -2π,β ],ε e -zζ ∨ ϕ (ζ )dζ + R β ,ε e -zζ ϕ(ζ )dζ .
From the arguments used in Sect. 7.6.1, we see that ϕ β defines an element of

A ≤0 ( ( β ). Moreover, if ∨ ψ is another major of ϕ which is summable in the direc- tion β (for instance ∨ ϕ - ∨ ψ ∈ O(C) ≤1
), then its Laplace integral ψ β coincide with ϕ β as elements of A ≤0 ( ( β ). Thus ϕ β is independent of the chosen summable major and only depends on ϕ ∈ SING sum θ ,α . This allows us to write ϕ β = L β ϕ . Making β varying in J, the functions L β ϕ obviously glue together analytically (by Cauchy and using the independence of L β ϕ with respect to the chosen summable major), to give and element L J ϕ of Γ (J , A ≤0 ).

Definition 7.32. The morphism

L β : SING sum θ ,α → A ≤0 ( ( β ) is called the Laplace transform in the direction β ∈ J =] -α + θ , θ + α[. The morphism L J : SING sum θ ,α → Γ (J , A ≤0 ) is called the Laplace transform in the direction J =] -α + θ , θ + α[.
We recover with the following proposition the examples given in the introduction of the chapter, see also [18].

Proposition 7.12. The singularities I σ and Jσ,m belong to SING sum θ ,α for any direction θ and any α > 0. Moreover, for any direction β ∈ S 1 ,

L β I σ (z) = 1 z σ , L β Jσ,m (z) = (-1) m log m (z) z σ , z ∈ Π β 0 ⊂ C.
This has the following consequences:

Proposition 7.13. For all σ 1 , σ 2 ∈ C, for all m 1 , m 2 ∈ N I σ 1 * I σ 2 = I σ 1 +σ 2 and Jσ 1 ,m 1 * Jσ 2 ,m 2 = Jσ 1 +σ 2 ,m 1 +m 2 .
Proof. From proposition 7.12, we deduce that In definition 7.32, we meant morphisms of vector spaces. As a matter of fact, these are morphisms of C-differential algebras. This is the matter of the following proposition.

L I σ 1 = 1 z σ 1 and L I σ 2 = 1 z
∨ ϕ * Γ ×Γ ∨ ψ (ζ ) = 1 2iπ Γ ×Γ e ν(ζ -(ξ 1 +ξ 2 )) ζ -(ξ 1 + ξ 2 ) ∨ ϕ (ξ 1 ) ∨ ψ (ξ 2 )dξ 1 dξ 2 , (7.12)
(see 7.9), where Γ = Γ β ,ε,η 1 ,η 2 is as in definition 7. . In (7.12), ν ∈ C is a free parameter which can be chosen at our convenience. We now assume that ϕ , ψ ∈ SING sum θ ,α and that ∨ ϕ , ∨ ψ are summable majors in the direction β . In that case, choosing ν = |ν|e -iβ with |ν| large enough to ensure the integrability, one can rather consider the convolution product ϕ * ψ as represented by (7.12), but this time with an endless path Γ = Γ β ,ε (see definition 7.7). This construction gives a major of ϕ , ψ which is summable in the direction β . Moreover, the arguments used in the proof of the proposition 7.10 show that L β ϕ * ψ = L β ϕ L β ψ .

Example 7.7. We consider the formal Borel transform w (0,0) = B w (0,0) where w (0,0) is the formal series solution of the prepared ODE (3.6) associated with the first Painlevé equation. We know by theorem 3.3 that w (0,0) can be analytically continued to the star-shaped domain • R (0) with at most exponential growth of order 1 at infinity along non-horizontal directions. We set w (0,0) = w (0,0) ∈ SING int . Then w (0,0) ∈ SING sum π/2,π/2 (or w (0,0) ∈ SING sum -π/2,π/2 ) : just consider the major

∨ w (0,0) (ζ ) = w (0,0) (ζ ) log(ζ )
2iπ . The Laplace transform L ]0,π[ w (0,0) is well-defined and gives a section of A ≤0 on ] -3π/2, π/2[. As a matter of fact,

L ]0,π[ w (0,0) = L ]0,π[ w (0,0) = S ]0,π[ w (0,0)
and L ]0,π[ w (0,0) can be thought of as belonging to the space of sections Γ (] -3π/2, π/2[, A 1 ). We now consider the singularity I σ * w (0,0) , for any σ ∈ C. Using propositions 7.12 and 7.14, this singularity belongs (for instance) to SING sum π/2,π/2

and L ]0,π[ I σ * w (0,0) = L ]0,π[ I σ L ]0,π[ w (0,0) = 1 z σ S ]0,π[ w (0,0) , this time viewed as a multivalued section A ≤0 on ] -3π/2, π/2[⊂ S 1 .

Spaces of resurgent functions

Preliminaries

We refer the reader to [1] (Pré I.3, lemme 3.0) for the proof of the following keylemma, the idea of which being due to Ecalle. Lemma 7.5. Let R 0 > 0 be a real positive number and Γ ⊂ C be an embedded curve, transverse to the circles |ζ | = R for all R ≥ R 0 . Let Φ be a holomorphic function on a neighbourhood of Γ . Then, for any continuous function m : R + → R + so that inf{m([0, ξ ])} > 0 for all ξ > 0, there exists

Ψ ∈ O(C) such that, for all ζ ∈ Γ , |Φ(ζ ) +Ψ (ζ )| ≤ m(|ζ |).
In what follows, we use the notations introduced in definition 7.7. We also recall that C \ Z stands for the universal covering of C \ Z. One may also think of C \ Z as the universal covering of C \ θ =πk, k∈Z {me iθ | m ∈ N }. Lemma 7.6. Let ϕ ∈ SING be a singularity which can be determined by a major analytically continuable to C \ Z. Then, for any direction θ and any ε > 0 small enough, the singularity ϕ has a major |η|) for all η ∈ Γ I,ε . One can also assume that |H 0 (s)| is bounded and these conditions ensure the integrability for the integral

∈ O(C) so that Φ 2 = Φ 1 +Ψ 1 satisfies |Φ 2 (η)| ≤ exp(-
Φ(ζ ) = 1 2iπ H 0 Φ 2 (η) ζ -η
dη which thus, defines a holomorphic function on

• S ε (I).
Moreover, one easily sees by Cauchy that Φ = Φ 2 +Ψ 2 where

Ψ 2 ∈ O 0 . One ob- serves that |ζ -η| ≥ ε -ε for (ζ , η) ∈ • S ε (I)×Γ I,ε , with ε > ε. Thus Φ is bounded on • S ε (I).
Notice that Φ 2 inherits from Φ 1 the property of being analytically continuable to C \ Z. Thus one can analytically continue Φ to C \ Z by Cauchy, by deformation of the contour by isotopies 5

H : (s,t) ∈ R × [0, 1] → H(s,t) = H t (s) ∈ C \ Z that
C \ [0, e iθ ∞[= • ∞ 0 ( Ǐ), Ǐ =] -2π + θ , θ [; 2. |Φ(η)| ≤ exp(-|η|) for all η ∈ Γ I,ε , where Γ I,ε = -∂ • S ε (I) ⊂ • ∞ 0 ( Ǐ); 3. Φ can be analytically continued to C \ Z.
Proof. Just consider first the function Φ 1 given by lemma 7.6, then use lemma 7.5 to define Φ from Φ 1 .

The above lemmas 7.6 and 7.7 motivate the introduction of new Riemann surfaces that will be used in a moment.

, γ 2 ∈ R θ ,ζ 0 , γ 1 (θ ,ζ 0 )
∼ γ 2 when one of the following conditions is satisfied: either γ 1 ∼ • R θ ,(0) γ 2 or γ 1 ∼ C\Z γ 2 ; or else there exists γ

3 ∈ A θ ,ζ 0 ∩ B ζ 0 such that γ 1 ∼ • R θ ,(0) γ 3 γ 2 ∼ C\Z γ 3 or γ 1 ∼ C\Z γ 3 γ 2 ∼ • R θ ,(0) γ 3 .
Let γ be an element of R θ ,ζ 0 . We denote by cl θ ,ζ 0 (γ) its equivalence class for the relation (θ ,ζ 0 ∼ . We finally set: Throughout this course we usually simply write "resurgent functions" or "resurgent series" instead of Z-resurgent functions or Z-resurgent series, since there is no risk of misunderstanding.

R θ Z,ζ 0 = {cl θ ,ζ 0 (γ) | γ ∈ R θ ,ζ 0 } and p θ ,ζ 0 : cl θ ,ζ 0 (γ) → γ(1) ∈ • R θ ,( 0

Resurgent singularities and convolution

Theorem 7.1. The space RES is a C-differential commutative and associative convolution algebra with unit δ , and CONS ⊂ RES is a subalgebra. Therefore, the space RES is a C-differential commutative and associative algebra and CONS ⊂ RES is a subalgebra.

Proof. (Adapted from [8,1]. The reader should look before at the reasoning made for the proof of proposition 4.6).

It is enough to only show that RES is a convolution space. We take two singularities ϕ , ψ ∈ RES, we choose a direction θ and we suppose 0 < ε 1.

By lemma 7.7, resp. lemma 7.6, ϕ , resp. ψ , has a major such that its restriction as a sectorial germ of codirection ) , that can be analytically continued to the Riemann surface (R θ , p) and moreover, satisfies the condition:

I =] -π/2 + θ , θ + π/2[, can be represented by a function ∨ ϕ , resp. ∨ ψ , holomorphic on • R θ ,( 0
1. | ∨ ϕ (η)| ≤ exp(-|η|) for all η ∈ Γ I,ε , where Γ I,ε = -∂ • S ε (I) ⊂ • R θ ,(0) ; 2. ∨ ψ is bounded on • S ε (I).
We know by lemma 7.1 that ζ -Γ I,ε ⊂ • S ε (I) for every ζ ∈ • S 2ε (I). We also think of Γ I,ε as an embedded curve H 0 : R → C with |H 0 (s)| bounded. Therefore, the above properties and the dominated Lebesgue theorem, ensure that the integral

χ(ζ ) = ∨ ϕ * H 0 ∨ ψ (ζ ) = H 0 ∨ ϕ η ∨ ψ (ζ -η)dη (7.
+ ξ ) = H 0 ∨ ϕ η ∨ ψ (ξ + ζ 0 -η)dη.
We take a smooth path γ : [0, 1] → C \ Z starting from ζ 0 = γ(0). We fix R ε so that γ([0, 1]) ⊂ D(0, R) and length(γ) < R. We will get the analytic continuation of χ along γ by continuously deforming

H 0 through an isotopy H : (s,t) ∈ R × [0, 1] → H t (s) ∈ C \ Z which is equal to the identity for |s| large enough. We pick a C 1 function η : C → [0, 1] satisfying {ζ ∈ C | η(ζ ) = 0} = Z.
We also set a C 1 function ρ : C → [0, 1] with compact support so that the conditions ρ| D(0,5R) = 1 and ρ| C\D(0,6R) = 0 are fulfilled. In what follows, we see H 0 as an embedded curve R → C and there is no loss of generality in supposing the existence of s 0 > 0 so that H 0 (s) ∈ D(0, 3R) for |s| < s 0 , else H 0 (s) ∈ C \ D(0, 3R). One considers the non-autonomous vector field [18], we can observe the following properties, for every t ∈ [0, 1]:

X(ζ ,t) = η(ζ )ρ(ζ ) η(ζ )+η γ(t)-ζ γ (t). We denote by g : (t 0 ,t, ζ 0 ) ∈ [0, 1] 2 × C → g(t 0 ,t, ζ 0 ) = g t 0 ,t (ζ 0 ) ∈ C the (well-defined global) flow of the vector field, that is t ∈ [0, 1] → ζ (t) = g t 0 ,t (ζ 0 )
1. φ t (γ(0)) = γ(t), that is γ is an integral curve. (Notice that ρ γ(t) = 1 because γ([0, 1]) ⊂ D(0, R)). 2. φ t (C \ Z) ⊂ C \ Z. (One has φ t (ω) = ω for any ω ∈ Z since η(ω) = 0). 3. φ t (ζ ) = ζ for any ζ ∈ C \ D(0, 6R) (since ρ| C\D(0,6R) = 0). 4. for every ζ ∈ D(0, 3R), φ t γ(0) -ζ = γ(t) -φ t ζ . Indeed, if t → ζ (t)
H 0 ⊂ • R θ ,( 0
) can be lifted uniquely with respect to p on the principal sheet R θ ,(0) of R θ . We note H 0 this lifting. We can use the lifting theorem for homotopies [11,5] to get the continuous mapping H

: (s,t) ∈ R × [0, 1] → H (s,t) = H t (s) ∈ R θ which makes commuting the following diagram: R θ H ↓ p R × [0, 1] -→ C. H We now set K : (s,t) ∈ R × [0, 1] → K(s,t) = K t (s) = γ(t) -H t (s). We know that K 0 (s) = γ(0) -H 0 (s) ∈ • S ε (I) ⊂ • R θ ,(0)
for every s ∈ R. In particular, one can lift K 0 uniquely with respect to p into an embedded curve K 0 on the principal sheet R θ ,(0) of R θ . Moreover K 0 (s) ∈ C \ Z, for every s ∈ R. Property 5 ensures that K t (s) stays in • S ε (I) for |s| ≥ s 0 , otherwise by property 4, K t (s) belongs to C \ Z. This implies that K t can be lifted uniquely with respect to p into an embedded curve K t which lies on the principal sheet R θ ,(0) of R θ for |s| ≥ s 0 . Applying again the lifting theorem for homotopies, one obtains a continuous mapping K (0) . With these notations, the germ of holomorphic functions χ at ζ 0 = γ(0) reads

: (s,t) ∈ R × [0, 1] → H (s,t) = H t (s) ∈ R θ that makes commuting the follow- ing diagram: R θ K ↓ p R × [0, 1] -→ C. K We finally introduce the two holomorphic functions Φ,Ψ ∈ O(R θ ) such that Φ(ζ ) = ∨ ϕ p(ζ ) , Ψ (ζ ) = ∨ ψ p(ζ ) for ζ ∈ R θ ,
χ(γ(0) + ξ ) = R Φ H 0 (s) Ψ (ξ + K 0 (s))H 0 (s)ds
and its analytic continuation along γ is obtained by

χ(γ(t) + ξ ) = R Φ H t (s) Ψ (ξ + K t (s))H t (s)ds. ( 7
| ∨ ϕ H t (s) | ≤ exp(-|H t (s)|). Also, for |s| ≥ s 0 , Ψ ξ + K t (s) = ∨ ψ K t (s) which is bounded since K t (s) ∈ • S ε (I).
Thus the integral (7.15) is well-defined. The fact that (7.15) provides the analytic continuations comes from the Cauchy formula, see analogous arguments in [18].

Supplements

One often uses other spaces in practice as we now exemplify.

The space RES (θ ,α) (L) The space R(θ,α) (L) was introduced by definition 4.24 and we know by proposition 4.6 that Cδ ⊕ R(θ,α) (L) is a convolution algebra. The following definition thus makes sense. Definition 7.40. We denote by R (θ ,α) (L) ⊃ R the C-differential commutative and associative convolution algebra made of singularities of the form ϕ = aδ + ϕ ∈ SING with ϕ ∈ R(θ,α) (L). The associated space of formal series is denoted by R (θ ,α) (L).

By its very definition, any element ϕ ∈ R(θ,α) (L) is a germ of holomorphic functions at 0 that can be analytically continued to the Riemann surface R (θ ,α) (L). This means that any ϕ ∈ R (θ ,α) (L) is a simple singularity that has a major ∨ ϕ which can be analytically continued to the universal covering R (θ ,α) (L) \ {0} of R (θ ,α) (L) \ {0}.

Since R (θ ,α) (L) is a convolution algebra, we know that for any two singularities ϕ , ψ ∈ R (θ ,α) (L), their convolution product ϕ * ψ belongs to R (θ ,α) (L) as well, thus has a major that can be analytically continued to R (θ ,α) (L) \ {0}. In substance, this comes from the property that ϕ * ϕ = ( ϕ * ϕ) for two integrable singularities (proposition 7.6). Now, what about the convolution product ϕ * ψ of two singularities ψ ∈ R (θ ,α) (L) and ϕ ∈ RES ? To give the answer, we prefer to shift to a more general case and we introduce a new definition. Definition 7.41. Let be θ ∈ {0, π} ⊂ S 1 , α ∈]0, π/2] and L > 0. We denote by RES (θ ,α) (L) the space made of singularities that have majors that can be analytically continued to the Riemann surface R (θ ,α) (L) \ {0}. The associated space of asymptotic classes is denoted by RES (θ ,α) (L) ⊂ ASYMP.

Proposition 7.16. The space RES (θ ,α) (L) is a C-differential commutative and associative convolution algebra with unit δ , contained RES and R (θ ,α) (L) as subalgebras.

Proof. The proof follows that of theorem 7.16 but for the fact that one adds the arguments used at the end of the proof of proposition 4.6.

The spaces RES (k) The spaces R (k) were introduced by definition 4.20. They provide new spaces of singularities which are worthy of attention. Definition 7.42. For k ∈ N , we denote by R (k) the space of singularities of the form ϕ = aδ + ϕ ∈ SING with ϕ ∈ R (k) . The associated space of formal series is denoted by R (k) . Remark 7.6. Notice that the set of spaces ( R (k) ) k∈N provides an inverse system of spaces whose inverse limit lim k) is R. This is why we sometimes write

← R (k) = k R (
R (∞) = R.
The space R (1) is of particular interest since, from propositions 4.1 and 7.6, R (1) makes a convolution algebra.

The space R (k) is made of simple singularities that have majors that can be analytically continued to the universal covering R (k) \ {0} of R (k) \ {0}. We now consider larger spaces of singularities.

Definition 7.43. Let k ∈ N be a positive integer. We denote by RES (k) the space of singularities that have majors that can be analytically continued to the Riemann surface R (k) \ {0}. We denote by RES (k) ⊂ ASYMP the space of asymptotic classes whose formal Borel transform belongs to RES (k) .

Remark 7.7. Notice again that lim

← RES (k) = k RES (k)
= RES, and we sometimes write RES (∞) = RES.

We will have a special interest in RES (1) because of the following analogous to proposition 7.16.

Proposition 7.17. The space RES (1) is a C-differential commutative and associative convolution algebra with unit δ . It contains RES and R (1) as subalgebras.

We omit the (rather lengthy) proof of this proposition. The main idea is to consider the integral representation (7.14) used in the proof of theorem 7.1 and to adapt the construction made in Sect. 4.3.

Conjecture 7.1. We conjecture that any space RES (k) makes a convolution algebra as well.

Alien operators

Alien operators are powerful tools for analysing the singularities of resurgent functions. These operators are carefully defined and discussed in [18], especially when they operate on the algebra Cδ ⊕ R simp of simple resurgent functions. Most of the arguments there can be easily adapted for alien operators acting on RES Z , once the study of singularities had been made. This is why we introduce the alien operators in a rather sketchy manner in what follows. We now consider two directions θ 1 , θ 2 ∈ S 1 and a singularity ϕ ∈ RES Z . Thinking of ϕ as a singularity of SING θ 1 ,α 1 (for some α 1 > 0), its minor ϕ can be seen as representing a sectorial germ ϕ ∈ O 0

A γ ω (θ 2 , θ 1 ) ∨ ϕ ∈ O 0 θ 2 the sectorial germ of direction θ 2 represented by ∨ ψ (ξ ) = cont γ ∨ ϕ (ω + ξ ) for ξ ∈ • 0 (I 2 ).
A γ ω (θ 2 ,θ 1 ) -→ RES L ↓↑ B L ↓↑ B RES A γ ω (θ 2 ,θ 1 ) -→ RES θ 1 = π(θ 1
). The very definition of RES (1) and the above reasoning lead straight to the following linear operators, for any integer m 1 ≥ 2 and any ε ∈ {-, +}:

A γ • θ 1 () ω 1 (θ 1 , θ 1 ) : RES (1) → SING θ 1 ,π , A γ • θ 1 (ε) m 1 -1 m 1 ω 1 (θ 1 , θ 1 ) : RES (1) → SING θ 1 +π/2,π/2
(7.16) We move to the next case k = 2, that is we consider the space RES (2) ⊂ RES (1) . Of course the above operators (7.16) still act on RES (2) but, however, their ranges can be made more precise. By the very definition of RES (2) , the minor ϕ of any singularity ϕ ∈ RES (2) , when considered as a sectorial germ, can be analytically continued along any path γ of type γ

• θ 1 ε n 1 with ε n 1 ∈ { (±) n 1 , (+) m 1 -1 , (±) n 1 , (-) m 1 -1 | (n 1 , m 1 ) ∈ (N ) 2 }. Moreover, introducing • θ 2 = • θ 1 + (n -1)π, ω 1 = e i • θ 1 , and ω 2 -ω 1 = e i • θ 2
, the analytic continuation cont γ ϕ of ϕ along γ is a germ of holomorphic functions whic can be analytically continued onto the simply connected domain

p(R ε n 1 , • θ ) = C \ {] -∞, p] ∪ [p + 1, +∞[} where ]p, (p + 1)[=]ω 1 , ω 2 [ when m 1 = 1, ]p, (p + 1)[=](m 1 -1)ω 2 , m 1 ω 2 [ when m 1 ≥ 2. Considering only odd values for n 1 (thus • θ 2 =
• θ 1 on S 1 ), one immediately sees that (7.16) becomes: (2) → RES (1) , (7.17)

A γ • θ 1 () ω 1 (θ 1 , θ 1 ) : RES
A γ • θ 1 (ε) 1 2ω 1 (θ 1 , θ 1 ) : RES (2) → SING θ 1 ,π A γ • θ 1 (ε) m 1 -1 m 1 ω 1 (θ 1 , θ 1 ) : RES (2) → SING θ 1 +π/2,π/2 , m 1 ≥ 3.
Notice in particular that the operator

A γ • θ 1 ()
ω 1 (θ 2 , θ 1 ) now acts on RES (2) as well, for any direction θ 2 ∈ S 1 .

The reasoning generalizes and we give the result.

γ • θ 1 (ε) m 1 -1 m 1 ω 1 (θ 1 , θ 1
) is well defined on RES (k) with the range:

A γ • θ 1 (ε) m 1 -1 m 1 ω 1 (θ 1 , θ 1 ) : RES (k) → RES (k-m 1 ) , 1 ≤ m 1 ≤ k -1 (7.18) A γ • θ 1 (ε) m 1 -1 m 1 ω 1 (θ 1 , θ 1 ) : RES (k) → SING θ 1 ,π , m 1 = k A γ • θ 1 (ε) m 1 -1 m 1 ω 1 (θ 1 , θ 1 ) : RES (k) → SING θ 1 +π/2,π/2 , m 1 ≥ k + 1.

Miscellaneous properties

We start with a simple result which is a consequence of the very definitions.

Proposition 7.18. For any alien operator of the form A γ ω (θ 2 , θ 1 ) : RES Z → RES Z , acting on RES Z , RES ( • θ ,α) (L) or RES (k) , for any singularity ϕ :

A γ ω (θ 2 , θ 1 ) ∂ ϕ = ( ∂ -ω)A γ ω (θ 2 , θ 1 ) ϕ . (7.19) In other words, [A γ ω (θ 2 , θ 1 ), ∂ ] = -ωA γ ω (θ 2 , θ 1
). We introduce new definitions before keeping on. Definition 7.48. For any k ∈ Z, one denotes by ρ k ∈ Aut(π) the deck transformation of the cover ( C, π), defined by: ρ The alien operators associated with a triple satisfy some identities as can be easily observed:

k : ζ = re iθ ∈ C → ρ k (ζ ) = re iθ +2iπk ∈ C.
γ ω (θ 2 , θ 1 ) : RES Z → RES Z , A γ ω (θ 2 , θ 1 + 2πk) = A γ ω (θ 2 , θ 1 )ρ k and A γ ω (θ 2 + 2πk, θ 1 ) = ρ -k .A γ ω (θ 2 , θ 1 ), for any k ∈ Z.
Let us consider a point ω ∈ Z and a given triple (γ, θ 1 , θ 2 ). One can extend the path γ into the path γλ k ω where λ k ω is a closed path near ω that surrounds that point like on Fig. 7.10, with winding number wind ω (λ k ω ) = k ∈ Z at that point. One can as well consider the path λ k 0 γ where λ k 0 is a closed path surrounding the origin with winding number wind ω (λ k 0 ) = k ∈ Z. A little thought provides the following result.

Proposition 7.20. We consider a triple (γ, θ 1 , θ 2 ) defining alien operator A γ ω (θ 2 , θ 1 ) : RES Z → RES Z at ω. We assume that γλ k ω , resp. λ k 0 γ is a product of paths so that λ k ω , resp. λ k 0 , is a closed path surrounding ω,resp. 0, and close to that point, with winding number wind ω (λ

k ω ) = k, resp. wind 0 (λ k 0 ) = k, k ∈ Z. Then, A λ k 0 γ ω (θ 2 , θ 1 ) = A γ ω (θ 2 , θ 1 )ρ k . , A γλ k ω ω (θ 2 , θ 1 ) = ρ k .A γ ω (θ 2 , θ 1 ). (7.20)
In particular,

A γ ω (θ 2 , θ 1 +2πk) = A λ k 0 γ ω (θ 2 , θ 1 ), A γ ω (θ 2 +2πk, θ 1 ) = A γλ -k ω ω (θ 2 , θ 1 ). (7.21)
We end with the following property.

Composition of alien operators

Alien operators on RES Z

The following definition is adapted from [18]. Definition 7.49. One calls alien operator at ω ∈ Z associated with the couple (θ 1 1 , θ m 2 ) any linear combination of composite operators of the form

θ θ θ ζ ζ ζ -2+ ζ -2+ γ -2+ λ - λ + γ 0 1 -2 -1 2
Fig. 7.12 The paths Γ

+ = γ 1 λ + ω 1 (ω 1 + γ 2 ) and Γ + = γ 1 λ -(ω 1 + γ 2 ), ω 1 = -2. A γ 2 ω 2 -ω 1 (θ 2 2 , θ 2 1 + 2πk) • A γ 1 ω 1 (θ 1 2 , θ 1 1 ) = A Γ + k ω 2 (θ 2 2 , θ 1 1 ) -A Γ - k ω 2 (θ 2 2 , θ 1 1 )
.

with Γ + k = γ 1 λ k ω 1 λ + ω 1 (ω 1 + γ 2 ) and Γ - k = γ 1 λ k ω 1 λ - ω 1 (ω 1 + γ 2 ) respectively, where λ k ω 1
stands for a closed path around ω 1 = -2 with winding number wind ω 1 (λ k ω 1 ) = k at that point.

What have been done in the above example can be generalized. This is the matter of the next proposition. Proposition 7.22. We consider the two alien operators A

γ 1 ω 1 (θ 1 2 , θ 1 1 ), A γ 2 ω 2 -ω 1 (θ 2 2 , θ 2
1 ) and we assume that θ

2 1 -θ 1 2 ∈ [0, 2π[. Then, for any k ∈ Z, A γ 2 ω 2 -ω 1 (θ 2 2 , θ 2 1 + 2πk) • A γ 1 ω 1 (θ 1 2 , θ 1 1 ) = A Γ + k ω 2 (θ 2 2 , θ 1 1 ) -A Γ - k ω 2 (θ 2 2 , θ 1 1 )
.

with Γ + k = γ 1 λ k ω 1 λ + ω 1 (ω 1 + γ 2 ) and Γ - k = γ 1 λ k ω 1 λ - ω 1 (ω 1 + γ 2 ) respectively, where λ k ω 1
stands for a closed path around ω 1 with winding number wind ω 1 (λ k ω 1 ) = k at that point, whereas λ + ω 1 and λ - ω 1 follows small arcs so that (λ - ω 1 ) -1 λ + ω 1 makes a loop around ω 1 counterclockwise. As a consequence, any alien operator at a point ω ∈ Z associated with the couple (θ 1 , θ 2 ) can be written as a linear combination of alien operators at ω associated with triples of the form (γ, θ 1 , θ 2 ).

We now focus on paths of type γ

• θ ε n . For m ∈ N , we take a (m -1)-tuple of signs ε = (ε 1 , • • • , ε m-1 ) ∈ {+, -} m-1 and n = (n 1 , • • • , n m-1 ) ∈ (N ) m-1 . We choose a direction θ 1 ∈ {πk, k ∈ Z}. Following definition 4.7, to a path of type γ • θ 1
ε n one associates a sequence of points and directions defined as follows :

• θ j+1 = • θ j + ε j (n j -1)π 1 ≤ j ≤ m -1 (7.23) ω j+1 -ω j = e i • θ j+1 0 ≤ j ≤ m -1 ω 0 = 0.
These data thus provide a uniquely defined alien operator

A γ • θ 1 ε n ω m (θ m , θ 1 ), once the direction θ m ∈ S 1 , • θ m = π(θ m ) is chosen. Theorem 7.2. Let m ∈ N be a positive integer, ε ∈ {+, -} m-1 , n ∈ (N ) m-1 and θ 1 ∈ {πk, k ∈ Z}. Let γ be a path of type γ • θ 1 ε n , ω m and
k ω k -ω k-1 (θ m , θ k ) • • • • • A γ 2 ω 2 -ω 1 (θ 2 , θ 2 ) • A γ 1 ω 1 (θ 1 , θ 1
) that satisfy the properties:

• (ω 1 , • • • , ω k ) ∈ Z k , k ∈ N and ω k = ω m ; • • θ m = • θ k ; • for every j = 1, • • • , k, the path γ j is of type γ • θ j (+) m j -1 , m j ∈ N ; • ∑ k j=1 m j ≤ m.
This theorem is of a purely geometric nature. We omit its proof (see [1] Sect. Rés II-2, see also [18,22]) and we rather produce two examples that explain the algorithm.

         • θ 1 = • θ 2 = 0 • θ 3 = • • • = • θ 6 = π ω 0 = 0, ω 1 -ω 0 = ω 2 -ω 1 = 1 ω 3 -ω 2 = • • • = ω 6 -ω 5 = -1. We set θ 1 = θ 2 = θ 1 = 0, θ 3 = • • • = θ 6 = θ 2 = π.
We start with the identity: ε n for ε = (+, -, +), n = (1, 3, 1) and

• θ 1 = 0. 0 1 3 4 2 A γ • θ 1 ((±) n l l-1 (ε) m l -1 ) m l ω l (θ l , θ 1 ) : RES (k) → RES (k-l-m l +1) , 1 ≤ m l ≤ k -l A γ • θ 1 ((±) n l l-1 (ε) m l -1 ) m l ω l (θ l , θ 1 ) : RES (k) → SING θ 1 ,π , m l = k -l + 1 A γ • θ 1 ((±) n l l-1 (ε) m l -1 ) m l ω l (θ l , θ 1 ) : RES (k) → SING θ 1 +π/2,π/2 , m 1 ≥ k -l + 2. (7.24) Equivalently, A γ • θ l (ε) m l -1 θ 1 .
The same reasoning can be generalized and gives the proposition.

A quite similar (and even simpler) reasoning gives the next result. Proposition 7.25. Let be k ∈ N and ϕ ∈ RES (k) . We suppose that for any

θ 1 ∈ {πk, k ∈ Z} ⊂ S 1 and any n ∈ N k-1 , the singularity A γ • θ 1 ((±) n k-1 ) ω k (θ k , θ 1
) ϕ belongs to RES (1) , where ω k is given by (7.23). Then ϕ belongs to RES (k+1) .

We eventually use theorem 7.2 to reformulate proposition 7.25.

Corollary 7.1. Let k ∈ N be a positive integer and ϕ ∈ RES (k) . We suppose that

A γ k ω k -ω k-1 (θ k , θ k ) • • • • • A γ 2 ω 2 -ω 1 (θ 2 , θ 2 ) • A γ 1 ω 1 (θ 1 , θ 1
) ϕ belongs to RES (1) for any composite operator that satisfies the properties:

• for every j = 1, • • • , k, the path γ j is of type γ • θ j (+) m j -1 , m j ∈ N ; • ∑ k j=1 m j = k.
Then ϕ belongs to RES (k+1) .

Alien derivations

We now specialize our analysis to some particularly interesting alien operators.

7.8.3.1 Definitions Definition 7.50. Let be θ ∈ {πk, k ∈ Z} ⊂ S 1 , α ∈]0, π/2] and L > 0. We set

• θ = π(θ ) ∈ {0, π} ⊂ S 1 . Let be ω = me iθ ∈ C for m ∈ {1, • • • , L }, resp. m ∈ N .
The alien operators ∆ + ω and ∆ ω at ω,

∆ + ω , ∆ ω : RES ( • θ ,α) (L) → SING θ ,α , resp. ∆ + ω , ∆ ω : RES → RES,
are defined as follows:

∆ + ω ϕ = A γ • θ (+) m-1 • ω (θ , θ ) ϕ (7.26) ∆ ω ϕ = ∑ ε=(ε 1 ,••• ,ε m-1 )∈{+,-} m-1 p(ε)!q(ε)! m! A γ • θ ε • ω (θ , θ ) ϕ ,
where p(ε), resp. q(ε) = m -1p(ε), denotes the number of "+" signs, resp. "-" signs in the sequence ε. Proof. We give the proof for the identity (7.27) only, so as to exemplify the use of singularities. Moreover we work on the space R ( • θ ,α) (L).

∆ + ω ( ϕ * ψ ) = (∆ + ω ϕ ) * ψ + ∑ • ω 1 + • ω 2 = • ω ∆ + ω 1 ϕ * ∆ + ω 2 ψ + ϕ * ∆ + ω ψ (
The reader is invited to compare with the proof made in [18] for simple resurgent functions.

There is no loss of generality in assuming that ϕ = ϕ, ψ = ψ with ϕ, ψ ∈ R(θ,α) (L). By proposition 7.6 one has ϕ * ϕ = ( ϕ * ϕ), therefore we can use arguments developed in chapter 4 (see in particular the proof of theorem 4.1), which allow us some abuse of notations.

The analytic continuation of the convolution product ϕ * ψ along a path γ of type

γ • θ (+) m-1 = π(γ θ (+) m-1
), ending at ζ = ω +

• ξ 0 near ](m -1)e i

• θ , me i

• θ [, is the germ of holomorphic functions defined as follows:

(cont γ ϕ * ψ)(ω + • ξ ) = H 1 ϕ(η 1 ) ψ(η 2 + • ξ - • ξ 0 )+ • ξ - • ξ 0 0 ϕ(ζ +η) ψ( • ξ - • ξ 0 -η)dη.
Here H 1 (= p • H 1 ) is a symmatrically contractile path deduced from γ,

ϕ(η 1 ) = cont H 1 |[0,s] ϕ H 1 (s) , ψ(η 2 + • ξ - • ξ 0 ) = cont H -1 1 |[0,s] ψ H -1 1 (s) + • ξ -
• ξ 0 and ϕ(ζ + η) = cont H 1 ϕ H 1 (1) + η . To get the associated singularity, that is ∆ + ω ( ϕ * ψ ), one only needs to consider the restrictions: 1. of the first integral near the "pinching points" (see Fig. 7.14), where one easily recognizes convolution products for majors and these provide the contribution

∑ • ω 1 + • ω 2 = • ω ∆ + ω 1 ϕ * ∆ + ω 2
ψ to the singularity ∆ + ω ( ϕ * ψ ); 2. of the two integrals near the end points, which provide the missing contributions (use proposition 7.2).

This ends the proof.

Definition 7.52. The linear operators ∆ ω are called alien derivations and RES is called a resurgent algebra (since stable under alien derivations).

We refer to [18] (see also [1]) for the proof of the next statements. •

ω 1 ≺•••≺ • ω s ≺ • ω ∆ + ω-ω s-1 • • • • • ∆ + ω 2 -ω 1 • ∆ + ω 1 , (7.28)
∆ + ω = ∑ s∈N 1 s! ∑ arg(ω 1 )=•••=arg(ω s-1 )=θ 0≺ • ω 1 ≺•••≺ • ω s ≺ • ω ∆ ω-ω s-1 • • • • • ∆ ω 2 -ω 1 • ∆ ω 1 , (7.29)
In the above theorem, ≺ stands for the total order on [0, ω] induced by t ∈ [0, 1] → tω ∈ [0, ω].

The alien derivations own the property of generating the whole set of alien operators. We precise this claim with the following upshot from theorem 7.2 and theorem 7.4. (θ m , θ 1 ) at

• ω m associated with the triple (γ, θ 1 , θ m ) can be written as a Z-linear, resp. Q-linear combination of composite operators of the form

ρ k n . ∆ + ω n • • • • • ∆ + ω 2 • ∆ + ω 1 , resp. ρ k n . ∆ ω n • • • • • ∆ ω 2 • ∆ ω 1 ,
that satisfy the properties:

• ( • ω 1 , • • • , • ω n ) ∈ (Z ) n , n ∈ N and π ∑ n j=1 ω j = • ω m ; • • θ m = arg(ω n ) + 2πk n , k n ∈ Z; • ∑ n j=1 |ω j | ≤ m.
Example 7.13. We continue the example 7.10. The path γ is of type γ 0 ε for ε = (+, -, +) and we know that A γ 0 (+,-,+)

4 (0, 0) = ∆ + 4 -∆ + 2 • ∆ + 2 .
(On the right-hand side of the equality, (4, 2) stands for (4e i0 , 2e i0 )). Using theorem 7.4, one gets:

A γ 0 (+,-,+) 4 (0, 0) = ∆ 4 + 1 2! ∆ 3 • ∆ 1 + ∆ 2 • ∆ 2 + ∆ 1 • ∆ 3 + 1 3! ∆ 2 • ∆ 1 • ∆ 1 + ∆ 1 • ∆ 2 • ∆ 1 + ∆ 1 • ∆ 1 • ∆ 1 + 1 4! ∆ 1 • ∆ 1 • ∆ 1 • ∆ 1 -∆ 2 + 1 2! ∆ 1 • ∆ 1 • ∆ 2 + 1 2! ∆ 1 • ∆ 1 .
Example 7.14. We continue the example 7.11. The path γ is of type γ This can be expressed in term of alien derivatives as well.

We end with an observation. By its very definition, any singularity ϕ ∈ R (θ ,α) (L) has a regular minor. This property involves the following relationships for the action of the alien operators. (These are essentially consequences of propositions 7.19 and 7.20). Proposition 7.26. We suppose α ∈]0, π/2], L > 0 and m ∈ {1, • • • , L }. The following equalities hold for any k ∈ Z:

• for any ϕ ∈ R (0,α) (L), ∆ + me iπ2k ϕ = ρ -k . ∆ + me iπ0 ϕ , ∆ me iπ2k ϕ = ρ -k . ∆ me iπ0 ϕ ; • for any ϕ ∈ R (π,α) (L), ∆ + me iπ(2k+1) ϕ = ρ -k . ∆ + me iπ ϕ , ∆ me iπ(2k+1) ϕ = ρ -k . ∆ me iπ ϕ ; • moreover, if ϕ ∈ R (0,α) (L) ∩ R (π,α) (L) and if its minor ϕ is even, then ∆ + e iπ ϕ = ρ -1/2 . ∆ + 1 ϕ , ∆ e iπ ϕ = ρ -1/2 . ∆ 1 ϕ with 1 = e i0 , while if ϕ is odd, then ∆ + e iπ ϕ = -ρ -1/2 . ∆ + 1 ϕ , ∆ e iπ ϕ = -ρ -1/2 . ∆ 1 ϕ . We have already describe the action of the alien operators on the spaces RES (k) . We can draw some consequences from theorem 7.3. One can apply theorem 7.3 to get: ∆ ω ϕ * ψ = ∆ ω ϕ * ψ + ϕ * ∆ ω ψ ∈ SING θ ,α .

Also, we know that ∆ ω ϕ and ∆ ω ψ belong to RES (k-m) or SING θ ,π depending on |ω|. Finally when 1 ≤ |ω| ≤ k -1, one can work in RES (1) ⊃ RES (k-m) which is a convolution algebra by proposition 7.17 and this provides the conclusion. We add a property that will be useful in the sequel.

Corollary 7.3. Let k ∈ N be a positive integer and ϕ ∈ RES (k) . We suppose that for

any n ∈ N , ∆ ω n •• • ••∆ ω 2 •
∆ ω 1 ϕ belongs to RES (1) for any composite operator that satisfies the properties:

( • ω 1 , • • • ,
• ω n ) ∈ (Z ) n and ∑ n j=1 |ω j | = k. Then ϕ belongs to RES (k+1) .

Proof. This is a direct consequence of both corollary 7.1 and theorem 7.2.

Ramified resurgent functions

As already said, one uses various spaces of resurgent functions, accordingly to the problem under consideration. We introduce some of them.

Simple and simply ramified resurgent functions

We start with the resurgent algebra of simple resurgent singularities, much discussed in [18] (see also [1,7]) and we make use of proposition 7.6. The minimal polynomial of the action of ρ on ϕ is P(X) = ∏ i (Xω i ) r i with ∑ i r i = d + 1. Write:

P(X) = (X -ω 1 ) r 1 -1 ∏ i =1 (X -ω i ) r i = (X -ω i ) r 1 -1 Q(X).
From the fact that (ρω 1 ) P(ρ) ϕ = 0, we deduce the identity P(ρ) ϕ = φ z σ 1 with φ ∈ C[[z -1 ]] 1 and a convenient σ 1 ∈ C such that ω 1 = e -2iπσ 1 . Since

P(ρ) log r 1 -1 (z) z σ 1 = Q(ρ) (r 1 -1)! ω r 1 -1 1 z σ 1 = Q(ω 1 )(r 1 -1)! ω r 1 -1 1 z σ 1 ,
we see that necessarily P(ρ) J σ 1 ,r 1 -1 ϕ σ 1 ,r 1 -1 = (-1) r 1 -1 φ z σ 1 and

ϕ σ 1 ,r 1 -1 = (-1) r 1 -1 φ (r 1 -1)!ω r 1 -1 1 Q(ω 1 )
.

We finally observe that P(ρ) φ -J σ 1 ,r 1 -1 ϕ σ 1 ,r 1 -1 = 0 and we can apply the induction hypothesis on φ -J σ 1 ,r 1 -1 ϕ σ 1 ,r 1 -1 . This ends the proof.

We are in good position to define the ramified resurgent functions [24,7,8], see also [15]. We state a result that derives directly from lemma 7.10 U k e -λ λ λ .kz W k (z),

where W 0 = w 0 = w

[0] 0 and for any n ∈ N and any k ∈ Ξ n+1,0 \ Ξ n,0 ,

W k = n ∑ l=0 1 l! (κ κ κ.k) l log l (z) W [0] k-l , W [0] k = z -τ τ τ.k w [0] k . (8.2)
The formal series w 0 ∈ C[[z -1 ]] solves (3.6), namely

P(∂ ) w 0 + 1 z Q(∂ ) w 0 = F(z, w 0 ) = f 0 + f 1 w 0 + f 2 w 2 0 , (8.3)
P(∂ ) = ∂ 2 -1, Q(∂ ) = -3∂ , f 0 (z) = 392 625 z -2 , f 1 (z) = -4z -2 , f 2 (z) = 1 2 z -2 , while
the W k satisfy a hierarchy of equations given by lemma 5.3 that we recall: As a rule and apart from some integrable equations, the resurgence cofficients are seldom known by closed formulas but can be calculated numerically : see for instance [9] and specifically [26] for hyperasymptotic methods, see also [1]. For the first Painlevé equation and its Stokes cofficients, an explicit expression has been obtained by Kapaev [17,18,19] using isomonodromic methods, see also [33,20] for an exact WKB offspring. This result has also been founded by Costin et al. [8] by means of resurgent analysis and we give this expression. The Stokes coefficients are also known for the second Painlevé equation, see [15] and references therein. It is likely that the method of Costin et al. [8] can be used to get the other resurgence cofficients for the first Painlevé equation.

P k W k = ∑ k 1 +k 2 =k |k i |≥1 W k 1 W k 2 2! ∂ 2 F(z, w 0 ) ∂ w 2 , ( 8
We saw with corollary 5.2 that the formal integral can be interpreted as a formal transformation w = Φ(z, u), Φ(z, u) = ∑ k∈N 2 u k w

[0] k (z) ∈ C[[z -1 , u]
] that formally conjugates the prepared equation (3.6) to its normal form (5.66). We mentioned (without proof) in Sect. 6.3 that this formal transformation gives rise to analytic transformations through Borel-Laplace summation. In other words, equation (3.6) and the normal form (5.66) are analytically conjugated.

It can be shown (see for instance the arguments given in [3]) that for any two differential equations that are formally conjugated to (5.66), then these differential equations are analytically conjugated if and only if their resurgence coefficients are the same. Therefore in this way, the resurgence coefficients are also called the holomorphic invariants of Ecalle. See [11] for further details.

Stokes phenomenon

Knowing the Stokes coefficients A 0 (ω) provides a complete description for the lower order Stokes phenomenon. In what follows, we use the notations of theorem 8.1 and we denote θ j i = arg(ω j i ), i = 1, 2, j ∈ Z. We simply refer to [25] for the notion of "symbolic Stokes automorphism" ∆ / + θ j i and of "symbolic Stokes infinitesimal generator" ∆ / θ j i , for a given direction θ j i . We only recall their expressions and relationships, in our frame:

∆ / + θ j i , ∆ / θ j i : k∈N e -kλ i z R Z ram → k∈N e -kλ i z R Z ram , ∆ / + θ j i = Id + ∑ k 0 ∈N • ∆ + k 0 ω j i , ∆ / θ j i = ∑ k 0 ∈N • ∆ k 0 ω j i ∆ / + θ j i = exp ∆ / θ j i = Id + ∑ ∈N e -k 0 λ i z ! ∑ k 1 +•••+k =k 0 k i ≥1 ∆ k ω j i • • • • • ∆ k 1 ω j i . (8.8)
Let us consider the formal series w 0 . From theorem 8.1, one sees that ∆ / + θ j i w 0 = w 0 + ∑ k∈N A 0 (ω j i ) k e -kλ i z W ke i (8.9)

we can apply corollary 7.2 : for any ω ∈ C so that

• ω = ±1 (the so-called seen singularities), the alien derivative ∆ ω W k is well-defined. If these alien derivatives belong to RES (1) , then we see with corollary 7.3 that the singularities W k belongs to RES (2) . A reasoning by induction allows to conclude.

In this section, we explain how to calculate these alien derivatives with various methods and we direct our efforts towards w 0 .

Preparations

The formal series w 0 being solution of the equation (8.3), we introduce by proposition 7.4 the singularities w 0 = w 0 , f 0 = f 0 , f 1 = f 1 and f 2 = f 2 . Notice that f 0 , As already said, we know that w 0 belongs to RES (1) , resp. w 0 belongs to RES

f
, and corollary 7.2 can be applied : with the notations of theorem 8.1, W = ∆ ω 0 1 w 0 is a well-defined singularity of SING 0,π , resp. W = ∆ ω 0 1 w 0 is a well-defined asymptotic class of ASYMP 0,π .

The singularities f 0 , f 1 , f 2 and I 1 are all constant of resurgence. Therefore, they vanish under the action of any alien derivation. Adding to this remark the fact that For the same reasons, the asymptotic class W is solution in ASYMP 0,π of a linear ODE: The differential equation (8.16) is nothing but the equation P e 1 ( w 0 ) W = 0 (8.17

P(∂ -1) W + 1 z Q(∂ -1) W = ∂ F(z,
)
where P e 1 is the linear operator recalled in (8.4). We know by lemma 5.4 that the differential equation P e 1 ( w 0 ) W = 0, that is (8.17 We claim that the general solution of equation (8.17 As promised, we show proposition 8.1 by two different approaches in the sequel.

Alien derivations, first approach

We follow here ideas developed in [16,27], see also [30,29,23]. We start with the following results that come from general nonsense in 1-Gevrey theory and its proof is saved for an exercise. Remark 8.2. More general results along that line in resurgence theory can be obtained, see [3,25] and specially [32].

We are now ready to calculate the alien derivative W = ∆ ω 0 1 w 0 ∈ SING 0,π . We consider the 1-Gevrey Nilsson series W e 1 = z 3/2 w e 1 ∈ Nils 1 solution of (8.16) (more precisely its transform through the Taylor map), and its associated singularity W e 1 = I -3 2 * w e 1 ∈ SING, where w e 1 = δ + w e 1 . (Remember that w e 1 has 1 for its constant term). Since w e 1 is invertible in C[[z -1 ]], so does w e 1 in SING, its inverse being given by w * -1 e 1 = δ + ∑ n≥1 (-1) n w * n e 1 . Accordingly, W e 1 is invertible in SING and W * -1 e 1 = I 3 2 * w * -1 e 1 . We now introduce the singularity S∈ SING 0,π defined by W = S * W e 1 (8.19) and we want to show that S= A 0 (ω 0 1 )δ for some A 0 (ω 0 1 ) ∈ C. Plugging (8. 19) into (8.15), using the property that ∂ is a derivation in SING 0,π (cf. proposition 7.6) and that W e 1 solves (8.15), one easily gets for S the following equation: We introduce S0∈ SING 0,π given by the identity:

Therefore D e 2 ( w 0 ) w e 2 = 0 in ASYMP 0,π and thus P e 1 ( w 0 )e 2z W e 2 = 0. The key point if that e 2z W e 2 belongs to e 2z ASYMP 0,π which is a vector space in direct sum with ASYMP 0,π .

Putting things together, keeping the same notations, we see that the kernel of the linear differential operator P e 1 (w i,0 ) in the space of sectorial germs of direction I i is spanned by W e 1 ,i and e 2z W e 2 ,i .

We now go back to the asymptotic class W ∈ ASYMP 0,π that satisfies P e 1 ( w 0 ) W = 0. Considering a refinement of (I i ) if necessary, one can find for each subscript i a representative W i ∈ A ≤0 (I i ) of W and a 1-exponentially flat germ b i ∈ A ≤-1 (I i ) such that P e 1 (w 0,i )W i = b i . To get W i , we apply the usual variation of constants method. One gets W i under the form

W i = B i (z) +C 1 W e 1 ,i +C 2 e 2z W e 2 ,i , C 1 ,C 2 ∈ C, (8.23)
2B i (z) = W e 2 ,i z -3 W e 1 ,i .b i -W e i ,i z -3 W e 2 ,i .b i .

It is a simple exercise to show that B i belongs to A ≤-1 (I i ) and one easily concludes that W i has to be equal to C 1 W e 1 ,i modulo A ≤-1 (I i ).

Depending on the arc, the term C 2 e 2z W e 2 ,i either belongs to A ≤-1 (I i ) (so one can take C 2 = 0) or escapes from W i ∈ A ≤0 (I i ) (thus one has to impose C 2 = 0).

This ends the second proof of proposition 8.1: the general solution of the linear equation P e 1 ( w 0 ) W = 0 in ASYMP 0,π is C 1 W e 1 and, consequently, there exists a constant A 0 (ω 0 1 ) ∈ C so that ∆ ω 0 1 w 0 = A 0 (ω 0 1 ) W e 1 in ASYMP 0,π . Thus, ∆ ω 0 1 w 0 can be uniquely continued to an element of ASYMP. ∏ k∈Z e -kz ASYMP 0,π , are spanned by the independent solutions W e 1 ∈ ASYMP and e 2z W e 2 ∈ e 2z ASYMP, resp. their restrictions in ASYMP 0,π and e 2z ASYMP 0,π respectively. This result can be generalized as follows.

Conclusion

P k W = ∑ k 1 +k 2 =k |k i |≥1 W k 1 W k 2 2! ∂ 2 F(z, w 0 ) ∂ w 2 , P k = P k ( w 0 ), (8.24)
then there exist uniquely determined constants C 1 ,C 2 ∈ C so that W = W k +e λ λ λ .kz C 1 e -λ 1 z W e 1 +C 2 e -λ 2 z W e 2 . (8.25)

Proof. The general formal solution for the equation (8.4) is of the form W = W k + e λ λ λ .kz C 1 e -λ 1 z W e 1 +C 2 e -λ 2 z W e 2 . We already know that e λ λ λ .kz C 1 e -λ 1 z W e 1 +C 2 e -λ 2 z W e 2 provides the general solution for the homogeneous equation P k ( w 0 ) W = 0 in ∏ ∈Z e -z ASYMP θ ,α . This asymptotic class W k is of the form (8.2), namely

W k = ∑ l 1 l! (κ κ κ.k) l log l (z)z -τ τ τ.k w [0] k , w [0] k = w [0] k , with w [0] k ∈ C[[z -1
]] 1 satisfying as linear differential equation given in corollary 5.1. This allows to conclude that W k is a particular solution for the equation (8.24) and one ends the proof with the arguments of the above second approach.

A step further

What have been previously done works as well for the other alien derivatives

(proposition 8.1), one infers from corollary 7.3 that w 0 belongs to R (2) . In particular, the alien derivatives ∆ 2ω j 1 w 0 ∈ SING 2π j,π , resp. We can keep on that way to get the complete resurgent structure for w 0 and, at the same time, to analytically continue its minor w 0 . Let us see what happens a step further.

To show that w 0 belongs to R (3) , we have to complete the informations given by lemma 8.4. Following corollary 7.3, we would like to show that ∆ ω 2 • ∆ ω 1 w 0 belongs to RES (1) for any ω 1 , ω 2 ∈ C so that

Let us look at W = ∆ ω 0 1 W e 1 ∈ ASYMP 0,π . From the identity P e 1 ( w 0 ) W e 1 = 0 (equation (8.16)) and corollary 7.2, we draw:

P(∂ -2) W + 1 z Q(∂ -2) W = ∂ F(z, w 0 ) ∂ w W + W e 1 ∆ ω 0 1 w 0 ∂ 2 F(z, w 0 ) ∂ w 2 ,
that is .27) where A 0 (ω 0 1 ) is the resurgent constant given in lemma 8.1. Observe that the general formal solution for the equation P 2e 1 ( w 0 ) W = A 0 (ω 0 1 ) W 2 To the formal integral w(z, U), one associates its analogue through the mapping ram :

P 2e 1 ( w 0 ) W = A 0 (ω 0 1 ) W 2 e 1 ∂ 2 F(z, w 0 ) ∂ w 2 . ( 8
w (z, U) = ∑ k∈N 2 U k e -λ λ λ .kz W k , W k = ram W k . (8.32)
We pick ω ∈ C and we assume for the moment that To fix the idea, suppose that

• ω = k 0 λ 1 with k 0 = 1 at the moment. We get from the decomposition (8.31) the identity:

Definition 1 . 1 . 2 . 1 . 1 .

 11211 Let λ : [a, b] ⊂ R → C be a path starting at x 1 = λ (a) and ending at x 2 = λ (b). If u is a (germ of) holomorphic function(s) at x 1 which can be analytically continued along λ , we denote by cont λ u the resulting (germ of) holomorphic function(s) at x Remark Let O = x∈C O x be the set of all germs of holomorphic functions. We equip O with its usual Hausdorff topology, a basis B = {U (U, Φ)} of open sets being defined as follows: U (U, Φ) = {ϕ x ∈ O x | ϕ x germ of Φ at x ∈ U}, where U ⊂ C is a domain and Φ ∈ O(U). With the projection q :

Fig. 1 . 1 2 γ 1

 1121 Fig. 1.1 The elliptic curve L viewed as the Riemann surface of p = (4u 3g 2 ug 3 ) 1/2 . The homology classes of the cycles γ 1 and γ 2 drawn generate H 1 (L ; Z)

1 dq 4q 3 -g 2 q -g 3 , ω 2 = 2 e 3

 1333

  z provides what is known as the zero curvature condition (or Lax equation), namely ∂ A ∂ x -∂ B ∂ z = [B, A] where [B, A] = BA -AB stands for the commutator. Expliciting this condition, one recovers the first Painlevé equation under the form

  With the following choice for α and β ,

Fig. 2 . 1

 21 Fig. 2.1 Left hand side : approximate period lattices in each quadrants Q i of z-plane. Right hand side, their images in the x-plane through the transformation x → z defined by (2.6)-(2.7)

Theorem 2 . 4 .

 24 The first Painlevé equation (2.1) has:

Proposition 2 . 1 .

 21 Each tritruncated solution of the first Painlevé equation is holomorphic on a full sector of the form {x ∈ C | arg x ∈ I, |x| ≥ 0}, where I stands for the closure of an open arc I of length |I| = 8π/5. Moreover, each tritruncated solution can be analytically continued to a disc |x| < r 0 with r 0 > 0 small enough.

. 5 ) 3 . 1 .

 531 Definition The differential equation (3.5), which reads

Remark 3 . 2 . 1 .

 321 One infers from (3.7) that the series w diverges since obviously |a 2m | ≥ (m!) 2 |a 2 | for m ≥ 1. 2. The differential equation (3.6) can be written as a fixed point problem, w = N(w), N(w) = -F(z, w) -3 z w + w . On can consider the differential operator N as acting on the ring C

Definition 3 . 2 .Definition 3 . 3 .

 3233 We denote by S 1 the circle of directions about 0 of half-lines on C. We usually identify S 1 with R/2πZ. Let I =]α, β [⊂ S 1 be an open arc. Its length is denoted and defined by |I| = βα. Let I ⊂ S 1 be an open arc. For 0 ≤ r < R ≤ ∞, we denote by • R r (I) the domain defined by • R r (I) = {ζ = ξ e iθ ∈ C | θ ∈ I, r < ξ < R}. In particular • R 0 (I) (resp. • ∞ r (I)) is an open sector with vertex 0 (resp. ∞) and aperture I. One denotes by • ¯R 0 (I) (resp. • ¯∞ r (I)) the closure of • R 0 (I) (resp. • ∞ r (I)) in C = C \ {0}. We use abridged notations • for sectors, when R or r is unspecified. A sector • 0 (I ) (resp. • ∞ (I)) is said to be a proper subsector of • 0 (I) (resp. • ∞ (I))

Proposition 3 . 2 .

 32 For any open arc I ⊂ S 1 of length |I| ≤ π, there exists a solution w of (3.6), holomorphic in a domain of the form • ∞ (I), whose Poincaré asymptotics at infinity in every proper subsector of • ∞ (I), is given by the formal solution w given by proposition 3.1. Proposition 3.2 thus describes the minimal opening of sectors on which holomorphic solutions w asymptotic to w exist. Through the transformations (3.4), (2.6) and (2.7), these solutions w corresponds to holomorphic functions u solutions of the first Painlevé equation, defined on open sectors of aperture 4π/5 : we thus get a first insight towards the truncated solutions for the first Painlevé equation (theorem 2.4).

Definition 3 . 5 .

 35 Let b 0 δ + g(ζ ) and c 0 δ + h(ζ ) be two elements of Cδ ⊕ C[[ζ ]]. Their convolution product (b 0 δ + g) * (c 0 δ + h) is defined by

Proposition 3 . 5 .

 35 The formal series w(z) ∈ C[[z -1 ]] is solution of (3.6) if and only if its minor w(ζ ) ∈ C[[ζ ]] is solution of the following convolution equation:

Proposition 3 . 6 .

 36 The series ∑ l≥0 h l (ζ ) is well defined in C[[ζ ]] and is formally convergent to the unique formal solution w(ζ ) ∈ C[[ζ ]] of the convolution equation (3.10).

ρ = C \ {±[ 1 ,Definition 3 . 10 .

 1310 +∞[}. (See Fig. 3.1). Let f (ζ ) = ∑ l≥0 a l ζ l be an analytic function on the open disc D(0, r). One denote by | f | the function defines by | f |(ξ ) = ∑ l≥0 |a l |ξ l .

Fig. 3 . 1

 31 Fig. 3.1 The domain • R (0) ρ .

Lemma 3 . 3 .

 33 Let U be a domain star-shaped from 0. Suppose that f and g are two holomorphic functions on U and satisfy the conditions: for every ζ ∈ U, f (ζ ) ≤ F |ζ | and g(ζ ) ≤ G |ζ | with F, G positive continuous functions on R + . Then f * g is holomorphic on U and for every ζ ∈ U, f * g(ζ ) ≤ F * G |ζ | and ζ f * g(ζ ) ≤ |ζ | F * G |ζ | .

Definition 3 . 11 .

 311 Let U ⊂ C be an open set. We denote by O(U) the space of functions holomorphic in U and continuous on the closure U.

Lemma 3 . 7 .≤ 8 2 l e 8 ρ

 378 For every ρ ∈]0, 1[ and every l ∈ N, the majorant function H l (ξ ) is a polynomial which satisfies: for every ξ ∈ C, | H l (ξ )| |ξ | .

Theorem 3 . 3 .

 33 The formal solution w of the prepared equation (3.6) associated with the first Painlevé equation, is a 1-Gevrey series and satisfies the following properties: 1. its minor w is an odd series, convergent to a holomorphic function which can be analytically continued to a function (still denoted by w) holomorphic on the cut plane • R (0) ; 2. w has at most exponential growth of order 1 at infinity along non-horizontal directions. More precisely, for every ρ ∈]0, 1[, there exist A > 0 and τ > 0 such that, for every ζ ∈ • R (0) ρ , | w(ζ )| ≤ Ae τ|ζ | ; 3. moreover in the above upper bounds one can choose A = 16 and τ = 8 ρ .

  l≥0

8 2 l e 8 ρ R ≤ 16e 8 ρRemark 3 . 3 .

 8833 R . This normal convergence ensures the uniform convergence on any compact subset of • R (0) of the series ∑ l≥0 h l (ζ ), which thus defines a function holomorphic on • R (0) . However, proposition 3.6 tells use that the series ∑ l≥0 h l converges to the formal Borel transform w of the formal solution w of the ODE (3.6). Better estimates can easily be obtained, see corollary 3.1 and exercise 3.3.

3. 3 . 2 . 1

 321 Convolution algebra and uniform norm Definition 3.12. Let U = U R ⊂ C be an open neighbourdhood of the origin, bounded and star-shaped, R = sup ζ ∈U |ζ | the "radius" of U. We denote by O(U), +, ., * the convolution C-algebra (without unit) of functions continuous on U and holomorphic on U. We denote by M O(U) the maximal ideal of O(U) defined by M O(U) = { f ∈ O(U), f (0) = 0}. We set

0 f

 0 ν and of Cδ ⊕ O(U), . ν as well. For f , g ∈ O(U) we have, writing ζ = |ζ |e iθ ∈ U, Re -ν|ζ | f * g(ζ ) = Re -ν|ζ | |ζ | (se iθ )g (|ζ |s)e iθ e iθ ds = R |ζ | 0 f (se iθ )e -νs g (|ζ |s)e iθ e -ν(|ζ |-s) e iθ ds.

  s n g (|ζ |s)e iθ e -ν(|ζ |-s) ds

Lemma 3 . 8 .

 38 Let be a, b, c, d ≥ 0, N ∈ N and ( F n ) 0≤n≤N be a sequence of entire functions, real and positive on R + , with at most exponential growth of order 1 at infinity. Then, the convolution equation w = d + [a + bξ ] * w+c F 0 + solution in C[[ξ ]], whose sum converges to an entire function w d (ξ ) with at most exponential growth of order 1 at infinity. The function w d (ξ ) is real, positive and non-decreasing on R + and, for every ξ ∈ C, the mapping d → w d (ξ ) is continuous on R + . Proof. Obviously, (3.22) has a unique solution w d ∈ R + [[ξ ]]. Its formal Laplace transform, w d = L ( w d ) ∈ R + [[z -1 ]], solves the algebraic equation

. 24 )

 24 Then for every d ≥ 0, for every ζ ∈ U, | w(ζ )| ≤ w d (ξ ) with ξ = |ζ |, where w d ∈ O(C) is the holomorphic solution of the convolution equation (3.22).

∑ n=1 ξ 0 F

 0 ζ )| ≤ c F 0 (ξ ) + ξ 0 [a + b(ξr)] | w(re iθ )| dr + c N n (ξr)| w * n (re iθ )| dr. We notice from (3.24) that | w(0)| = f 0 (0) p(0) , while w d (0) = c F 0 (0) + d, where w d solves (3.22). Remark that | w(0)| ≤ c F 0 (0) by definition of c and by hypothesis on F 0 . First case. We assume w d (0) > | w(0)|. We want to show that | w(ζ )| < w d (ξ ) for ζ on the ray ζ = ξ e iθ ∈ U.

  by continuity of w and w d , one can find α > 0 such that | w (|ζ 2 |α)e iθ | > w d (|ζ 2 |α), but this this contradicts again the definition of ζ 2 . Therefore | w(ζ 2 )| = w d (|ζ 2 |).

Therefore | w(ζ 2 )

 2 | ≤ w d (ξ 2 )d and we get a contradiction. As a conclusion, for every d > 0, for every ζ ∈ U, | w(ζ )| ≤ w d (ξ ) with ξ = |ζ |. Second case. The case w d (0) = | w(0)| (thus, in particular, d = 0) is deduced from the above result. Indeed, for a given ζ ∈ U, one has by | w(ζ )| ≤ w d (ξ ) for every d > 0. Since the mapping d → w d (ξ ) is continuous on R + (cf. lemma 3.8), one gets the result by letting d → 0.

•R

  (0) . One turns to the Grönwall lemma to get upper bounds. Working in the star-shapeddomain • R (0) ρ , ρ ∈]0,1[, one sees by lemma 3.2, lemma 3.3 and the Grönwall lemma 3.9, that for every ζ ∈ • R (0) ρ , | w(ζ )| ≤ w(ξ), ξ = |ζ |, where w(ξ) solves the following convolution equation: 1 M ρ,(0)

Definition 3 . 13 .

 313 Let θ ∈ S 1 be a direction and I =]α, β [⊂ S 1 be an open arc. We denote by ( θ ⊂ S 1 the open arc defined by ( θ =] -π 2θ , -θ + π 2 [, and ( I = θ ∈I ( θ . We denote by Ī = [α, β ] the closure of I and by I =]β , -α[ the complex conjugate open arc. Definition 3.14. For a direction θ and τ ∈ R, we denote by • Π θ τ the following open half-plane, bisected by the half-line e -iθ R + : • Π θ τ = {z ∈ C, ℜ(ze iθ ) > τ}, of aperture ( θ . Let I ⊂ S 1 be an open arc of length |I| ≤ π and γ : I → R be a locally bounded function. The domain • D(I, γ) is defined by •

a 1 -Definition 3 . 15 .

 1315 Gevrey series subject to the following conditions: • there exists an open arc I ⊂ S 1 of length |I| ≤ π so that the minor g can be analytically continued to the open sector • ∞ 0 (I); • for every direction θ ∈ I, | g(ξ e iθ )| ≤ A(θ )e γ(θ )ξ , ξ > 0, where A : I → R + and γ : I → R are locally bounded functions. Then the family (S θ g) θ ∈I of Borel-Laplace sums determines a holomorphic function on the domain • D(I, γ), denoted by S I g. Under the conditions of proposition 3.10, g is said to be Borel-Laplace summable in the directions of I. The function S I g ∈ O • D(I, γ) is called the Borel-Laplace sum of g in direction I. Proposition 3.11. Let g(z) = ∑ n≥0 b n z n ∈ C[[z -1 ]] be a formal series, Borel-Laplace summable in the directions of I ⊂ S 1 , an open arc of length |I| ≤ π. Then its Borel-Laplace sum S I g ∈ O • D(I, γ) is 1-Gevrey asymptotic to g on • D(I, γ) : for any proper-subsector • ∞ • D(I, γ), there exist constants C > 0 and A > 0 such that for every N ∈ N and every z ∈ • CN!A N |z| -N . (3.26) In this proposition, the property 3.26 essentially characterizes the Borel-Laplace sum. Indeed, notice that the sectorial neighbourhood of infinity • D(I, γ) is of aperture ( I which satisfies π < | ( I | ≤ 2π, and one can draw the following consequence from the Watson lemma (see [16]): let • ∞ (I ) be any sector such that |I | > π and I ⊂ ( )) be a holomorphic function which is 1-Gevrey asymptotic to g on • ∞ (I ). Then f and S I g coincide on • ∞ (I ) ∩ • D(I, γ). We eventually ends this reminder with the following statement: Proposition 3.12. Let I ⊂ S 1 be an open arc of length |I| ≤ π and f (z), g(z) ∈ C[[z -1 ]] be Borel-Laplace summable formal series in the directions of I. Then f g and ∂ f are Borel-Laplace summable formal series in the directions of I and S I ( f g) = (S I f )(S I g), S I (∂ f ) = ∂ (S I f ).

3. 4 . 2

 42 Formal series solution and Borel-Laplace summation 3.4.2.1 Borel-Laplace summation We go back to the formal solution w of equation (3.6). Theorem 3.3 and corollary 3.1 have the following consequences: Corollary 3.2. The Borel transform w ∈ O( • R (0) ) of the formal solution w of equation (3.6) satisfies the following property. For every δ ∈]0, π 2 [, there exist A δ > 0 and τ δ > 0 so that for every ζ ∈ • ∞ 0 (]δ , πδ [), | w(ζ )| ≤ A δ e τ δ |ζ | . (3.27) Moreover one can choose A δ = 4.22, τ δ = 4.22 sin(δ ) . Proof. One can define δ = sin -1 (ρ) = arcsin(ρ) ∈]0, π 2 [, for any ρ ∈]0, 1[. From corollary 3.2 and the properties of the Borel-Laplace summation, we see that for every δ ∈]0, π 2 [, the Borel-Laplace sum S θ w of w in any direction θ ∈]δ , πδ [, is well-defined and holomorphic in the half-plane • ) . These holomorphic functions glue together to give the Borel-Laplace sum S ]δ ,π-δ [ w, holomorphic in the domain • D(]0, π[, τ) with τ : θ ∈]0, π[ → τ(θ ) = 4.22 sin(θ ) . (See Fig. 3.2 and exercise 3.4). Moreover, since w formally solves (3.6), its Borel-Laplace sum S ]0,π[ w is a solution of this equation which is 1-Gevrey asymptotic at infinity to w on • D(]0, π[, τ). Similarly, the formal series w is Borel-Laplace-summable in the directions of the interval ]π, 2π[. This provides the Borel-Laplace sum S ]π,2π[ w which belongs to O • D(]π, 2π[, τ) and is 1-Gevrey asymptotic to w on • D(]π, 2π[, τ).

Fig. 3 . 2

 32 Fig. 3.2 The (shaded) domain • D(]0, π[, τ) for τ(θ ) = 4.22 sin(θ ) .

Definition 3 . 16 .

 316 We denote by S r (θ ) the open half-strip S r (θ ) = s∈R + D(se iθ , r), for r > 0 and θ a direction.

ρProposition 3 . 14 .

 314 and, by theorem 3.3, there exist A > 0 and τ > 0 such that for every ζ ∈ S r (θ ), | w(ζ )| ≤ Ae τ|ζ | with sin(θ ) = r + ρ. Also, from corollary 3.1, one can choose A = 4.22, τ = 4.22 ρ . As a consequence, proposition 3.13 can be applied. The reader will easily adapt the previous considerations when the directions θ ∈]π, 2π[ are considered. We summarize what have been obtained. The 1-Gevrey series w ∈ C[[z -1 ]] 1 , solution of the prepared equation (3.6) associated with the first Painlevé equation, is Borel-Laplace summable in the directions of the arc I 0

.36) 3 . 4 . 2 . 4

 3424 Asymptotics and approximations By Stirling formula one has N! ∼ √ 2πN N+ 1 2 e -N for large N. Since for a given z = 0 the function N → N N e -N (r|z|) N reaches its minimal value at n = r|z|, it turns out from formula (3.30) that one can estimate the value of w tri,0 or w tri,1 from the truncated series expansion N ∑ k=0 a k z k with N = r|z| where . is the entire part. This gives rise to the summation to the least term.

For I ⊂ S 1

 1 an open arc and γ : I → R locally bounded, the domain D(I, γ) is sent onto T D(I j , τ) ⊂ C through the mapping T , and we set

Fig. 3 . 3

 33 Fig. 3.3 The shaded domain is the projection of (I 0 , τ), image by the transformation (3.37), of the domain D(I 0 , τ) drawn on Fig. 3.2 for τ(θ ) = 4.22 | sin(θ )| . The dash

  2iπ 5 j . The first Painlevé equation (2.1) has 5 tri-truncated solutions u tri, j (x), j = 0, • • • 4. The tri-truncated solution u tri, j (x) is holomorphic in • (I j , τ), a sectorial neighbourhood of infinity of aperture K j =] -11 5 π, -3 5 π[-4 5 jπ, and has in • (I j , τ) a Gevrey asymptotic expansion of order 4/5 which determined u tri, j (x) uniquely. Moreover, for every x ∈ • (I j , τ),

Exercices 3 . 1 .

 31 We consider an ordinary differential equation of the form P(∂ )w = G(z, w, w , ..., w (n-1) )

2 . 3 .

 23 Let be ρ ∈]0, 1[. We consider the (unique) entire function w solution of the convolution equation ρ w(ξ) = 392 625 + 7 * w(ξ)+ 1 2 * w * w(ξ). We denote by w(z) the inverse Borel transform of w. Show that w(z) ∈ O |z| > 203 25ρ (consider the discriminant locus). Show that for |z| > 203 25ρ , w(z) = 784 625 (ρz -7) + (ρz -7) 2 -784 625 1/2 -1 , w(z) = O(z -1 ) at infinity, and | w(z)| ≤ 784 625 1 |ρz-7| ≤ 28 25 . Show that | w(ξ)| ≤ 5684 625ρ e 203 25ρ |ξ | for every ξ ∈ C.

4 .

 4 Deduce that for every ρ ∈]0, 1[ and every ζ ∈

Fig. 3 . 4 R 8 .

 348 Fig. 3.4 The shaded domain is the projection of 2 (I 2 , τ), image of the domain D(I 2 , τ) by the conformal mapping (3.37), for τ(θ ) = 1.4 | sin(θ )| .The dash lines recall the sectors (2.9).

4. 2 . 2 . 1 Definition 4 . 1 . 0 ∼

 221410 The space R Z,ζ 0 Let U 0 be a connected and simply connected neighbourdhood of the origin in C and ζ 0 ∈ U 0 = U 0 \ {0}. We denote by A ζ 0 (resp. B ζ 0 ) the set of paths in U 0 (resp. C \ Z) originating from ζ 0 , endowed with the equivalence relation ∼ U 0 (resp. ∼ C\Z ) of homotopy of paths with fixed extremities. We set R ζ 0 = A ζ 0 ∪ B ζ 0 and denote by ζ the relation on R ζ 0 defined as follows.

Definition 4 . 2 .

 42 Let γ be an element of R ζ 0 . We denote by cl ζ 0 (γ) its equivalence class for the relation ζ 0 ∼. We set • R = C \ Z and we define:

. 1 )Notice that p - 1 ζ 0 ( 0 )Definition 4 . 3 .

 110043 is reduced to a single distinguished point, the equivalence class of any γ ∈ A ζ 0 ending at the origin, because U 0 is simply connected. One denotes by 0 ∈ R Z,ζ 0 the unique pre-image of 0 by p ζ 0 . Let be ζ ∈ • R, one denotes by ζ ∈ R Z,ζ 0 one of its pre-image if exists. For any ζ ∈ R Z,ζ 0 , one denotes by • ζ = p ζ 0 (ζ ) its projection by p ζ 0 . 4.2.2.2 The Riemann surface R Z,ζ 0 The topological space R Z,ζ 0 We endow R Z,ζ 0 with a topology, a basis B = {U } of open sets being given as follows. Let ζ be an element of R Z,ζ 0 and set ζ = p ζ 0 (ζ ).

Exercise 4 . 2 . 2 . 4 . 1 .

 42241 any connected and simply connected open neighbourhood of ζ . To (U, ζ ) we associate the set U ⊂ R Z,ζ 0 made of all ξ = cl ζ 0 (γ 1 γ 2 ) where γ 1 satisfies γ 1 ∼ C\Z γ and γ 2 is any path in U originating from ζ . Show the following properties (hint : see the classical construction of the universal covering of C \ Z [7, 3] and adapt the arguments): 1. B = {U } provides a Hausdorff topology on R Z,ζ 0 ; 2. the projection p ζ 0 is a continuous mapping and even, a local homeomorphism : for every U ∈ B, the mapping p ζ 0 | U → U = p ζ 0 (U ) is a homeomorphism. 3. R Z,ζ 0 is arc-connected and simply connected. The Riemann surface R Z,ζ 0 The following proposition is a direct consequence of the properties detailed in exercise 4.Proposition The space R Z,ζ 0 is a topologically separated space, arc-connected and simply connected The projection p ζ 0 makes R Z,ζ 0 an étalé space on • R. By pulling back by p ζ 0 the complex structure of C, the space R Z,ζ 0 becomes a Riemann surface with a uniquely defined distinguished point 0 = p -1

Definition 4 . 4 .

 44 The class of isomorphisms of the Riemann surfaces (R Z,ζ 0 , p ζ 0 , 0) is denoted by (R Z , p, 0). In this course we often use abridged notation R.

Fig. 4 . 2 AFig. 4 . 3 A

 4243 Fig. 4.2 A path of type γ θ ε for ε = (+, -, +) and θ = 0. 0 1 2 3 4 Fig. 4.3 A path of type γ θ ε n

2 - 1 . 5 •

 215 is the floor function), one observes that the discs D(m, |m|ρ) do not overlap when |m| ≤ M (ρ) + 1. Definition 4.11. Let be ρ ∈]0, 1 5 [. We denote by M (ρ) ∈ N the positive integer defined by M (ρ) = ρ -1 -1 For any integer m ∈ Z such that |m| ≤ M (ρ) + 1, we denote by D m = D(m, |m|ρ) the closed disc centered at m with radius |m|ρ, and D 0 = {0}. For any θ ∈ {0, π}, we denote by D θ ρ ⊂ C the closed subset defined by D θ ρ = tζ | t ∈ [1, +∞[, ζ ∈ D e iθ (M+1) 0≤m≤M (ρ) D e iθ m . We set • P θ ρ = C \ D θ ρ . We denote by • Rρ the domain defined by • Rρ . The domains • P θ ρ satisfy the following property, the proof of which being left as an exercise : Lemma 4.2. Let be ζ be an element of • P θ ρ . For every n ∈ [1, M (ρ)], the closed set ζ -D e iθ n = {ζξ | ξ ∈ D e iθ n } is a subset of • P θ ρ . Definition 4.12. Under the hypotheses of definition 4.11, for any integer m ∈ [0, M (ρ)] and θ ∈ {0, π}, we define:

ρ

  and this contradicts the assumption ζ ∈ • Q m,0 ρ . Definition 4.13. Under the hypotheses of definition 4.11, for any integer m ∈ [1, M (ρ)] and any θ ∈ {0, π}, we denote by D m,θ ρ ⊂ C the closed subset defined by

ρ

  . For any integer m > M (ρ), we set • P m,θ ρ = / 0. For ε = ± we denote by • P (ε) m ,θ ρ the domain • P (ε) m ,θ ρ = • P m,θ ρ ∩{ζ | εe iθ (ℑζ ) ≤ 0}. (See Fig. 4.6). Definition 4.14. Under the hypotheses of definition 4.11, for any θ ∈ {0, π}, ε = ± and m ∈ N, we denote by

Fig. 4 . 6

 46 Fig. 4.6 The domain • P 2,π ρ . The set • P (-) 2 ,π ρ lies below the real axis, the set • P (+) 2 ,π ρ lies above the real axis.

  λ

Definition 4 . 15 .

 415 With the above notations, with ε = ± and m ∈ [1, M (ρ)] an integer, one sets

•

  R ρ . In the same line, the following lemma is a consequence of lemmas 4.2 and 4.3. Lemma 4.4. Let be m ∈ [1, M (ρ)], θ ∈ {0, π}, ε = ± and let V be the closure of R (ε) m ,θ ρ \ R (0) ρ . For every ζ ∈ V , and every integer n ∈ [1, m], • ζ -D e iθ n is a subset of • R ρ and there exists an open set U ⊂ R (ε) m-n ,θ ρ such that U and • ζ -D e iθ n are p-homeomorphic.

•

  R ρ ⊂ • R (definition 4.11) can be thought of as a complete real 2-dimensional Riemannian manifold with smooth (C 1 ) boundary embedded in the 2-dimensional euclidean space. The following lemma thus makes sense. Lemma 4.5. Let X ⊆ • R ρ be any closed space with smooth(C 1 ) boundary. For every two points ζ 1 , ζ 2 ∈ X, there exists a geodesic in every homotopy class of curves from ζ 1 to ζ 2 in X, and this geodesic may be chosen as a shortest path in the homotopy class.

Fig. 4 . 8

 48 Fig. 4.8 Shortest curve for ζ in R

( 1 )

 1 ρ (definition 4.15) meets the requirement: Lemma 4.6. The closed, connected and simply connected space with smooth C 1boundary R(1)

Lemma 4 . 7 .

 47 For every ζ ∈ R (1) ρ , there exists a unique path Λ in R (1) ρ , originating from 0 and ending at ζ , such that Λ is a shortest path in its homotopy class and is parametrized by arc-length. Remark 4.2. It is easy to construct Λ by hand.

  First case: ζ belongs to R (0) ρ . Consider the curve λ , with its arc-length parametrization, starting from 0 which follows the segment [0, • ζ ] ⊂ • R (0) ρ . The path Λ is obtained by lifting λ from 0 with respect to p on R (1) ρ .

Definition 4 . 16 .

 416 Let ζ be an element of ζ ∈ R (1) ρ . The unique C 1 -path Λ in R (1) ρ

( 1 )

 1 ρ which is our frame in what follows.

4. 3 . 2

 32 Shortest symmetric (Z, ρ)-homotopy Definition 4.18. Let Λ be a path in R (1)

ρ

  instance, pick a point ζ in R (0) ρ and let λ be the smooth path which follows the segment [0, in the forward direction with a constant velocity. The path Λ in R (0) ρ deduced from λ by lifting from 0 with respect to p, is shortest-symmetric. Proposition 4.3. Let ζ be any given point in R (1) ρ . There exists a unique continuous map H : (s,t) ∈ I × I → H (s,t) ∈ R (1) ρ , I = [0, 1], which satisfies the following conditions:

Fig. 4 . 9

 49 Fig. 4.9 The shortest symmetrically contractile path for

Lemma 4 . 10 .

 410 Let be ζ ∈ R (1) ρ . Either ζ ∈ R (0) ρ and then leng ρ (ζ ) = | • ζ |, or

Proposition 4 . 4 .

 44 For any two elements f , g ∈ O(U ρ,L ), their convolution product belongs to O(U ρ,L ). In other words, Cδ ⊕ O(U ρ,L ) is a convolution algebra.

( 1 )

 1 ρ , | w(ζ )| ≤ w d leng ρ (ζ ) , where w d ∈ O(C) stands for the holomorphic solution of the following convolution equation:

  This shows that | w(ζ 2 )| ≤ w d (ξ 2 )d and we get a contradiction. Second case The case | w(0)| = w(0) (thus d = 0) is done by an argument of continuity already used in the proof of lemma 3.9.

4. 5

 5 Application to the first Painlevé equation 4.5.1 A step beyond Borel-Laplace summability

Theorem 4 . 2 .

 42 The formal solution w of the prepared equation (3.6) associated with the first Painlevé equation satisfies the following properties:

  ρ,(1) |q|(1). Moreover one can choose M ρ,(1) = 6 5ρ 3

  We use the Grönwall lemma 4.11 (with d = 0), which tells us that for every ζ ∈ R (1) ρ , | w(ζ )| ≤ w d (ξ ), ξ = leng ρ (ζ ), where w(ξ) solves the following convolution equation 1 M ρ,(1) w = | f 0 | + 3 + | f 1 | * w+| f 2 | * w * w (just use lemma 4.12). Moreover, one can choose M ρ,(1) = 6 5ρ 3 . We would like to get explicit estimates. We consider w as the Borel transform of the function w, solution of the second order algebraic equation, 1 M ρ,(1)

ρ 3 = 4 ρ 3

 343 {H ∈ O(U), H ≤ ρ 3 }. It is easy to show that the mapping N |B ρ 3 : H ∈ B ρ 3 → N(H) ∈ B ρ 3 is contractive (remember: ρ ∈]0, 1/5]). Therefore, from contraction mapping theorem, the fixed-point problem (4.8) has a unique solution H in B ρ 3 . In return we deduce that w is an entire function and | w(ξ)| ≤ 4e |ξ | for every ξ ∈ C (see lemma 3.5). One ends with lemma 4.10.

1 ( 1 ., 1 2 )

 112 definitions 4.7 and 4.9), for any m ∈ N and any direction θ ∈ {0, π} ⊂ S 1 . To fix our mind, we consider a path γ of type γ 0 (+) m-The analytic continuation of w along γ gives a germ cont γ w which can be represented by a function holomorphic on the open disc D( 2m-1 2 adherent to m. Writing f m (ζ ) = cont γ w(m + ζ ), we get a function f m holomorphic on D = D(-1 2 , 1 2 ). However, theorem 4.2 translates into the fact that f m can be analytically continued to a wider domain as a "multi-valued function". Precisely, pick a point ζ 0 = 1 2 e iθ 0 ∈ π -1 (-1 2 ) above -1 2 on the Riemann surface ( C, π) of the logarithm. Let D = D(ζ 0 , 1 2 ) ⊂ C be the

Fig. 4 . 3 0

 43 Fig. 4.10 Comparison of right and left Borel-Laplace summation for the direction θ = 0.

Question 4 . 6 .Question 4 . 7 .Question 4 . 8 .

 464748 Can we analytically continue each ∨ f m into an element of ANA ? The answer is "yes" but requires further effort and supplements to resurgence theory, given in chapter 7. Taking this for granted, to ∨ f m thus corresponds a singularity f m ∈ SING deduced from w through the action of the alien operator denoted by A γ,ζ 0 m in [10], or more precisely to ∆ + m . Can we describe more precisely the singularities f m ?This is of course the key-question for describing the Stokes phenomenon. Partly, the reply relies on the formal integral associated with equation (3.6), which is the matter of the next chapter 5. The final answer will be given in the last chapter 8 of this course, with the use of the alien derivations. In the same spirit: At this stage, can we compare the sums S ]-π,0[ w ∈ O(• D(]π, 0[, τ))and S ]0,π[ w ∈ O( • D(]0, π[, τ)) ? (See proposition 3.14.) In other words, are we able to analyze the Stokes phenomenon ? Formulated another way, we would like to compare the right Borel-Laplace summation S 0 + w and the left one S 0 -w. Look at Fig. 4.10 : we have chosen two directions θ + ∈]π, 0[ and θ -∈]π, 0[ closed to zero, so that the Borel-Laplace sums S θ + w(z) = λ + e -zζ w(ζ )dζ and S θ - w(z) = λ - e -zζ w(ζ )dζ can be compared on their nonempty common domain of definition • Π θ + τ(θ + ) ∩ • Π θτ(θ -) . The curve λ -1 -λ + can be seen as a chain on the Riemann sphere C = C ∪ {∞} running from ∞ to ∞ and avoiding the points Z ⊂ C. In other words, λ -1 -λ + represents a cycle for the 1-homology group H 1 ( • R ∪ {∞}, ∞) which is homologous to the sum ∑ m k=1 γ k +C m+1 (with m = 2 on Fig. 4.10

γ ke

  -zζ w(ζ )dζ are of the form e -kz W k (z) where W k stands for a formal series which only depends on the still unknown singularity f k . The remaining integral C m+1 e -zζ w(ζ )dζ provides an

Definition 4 . 23 .

 423 Let be θ ∈ {0, π}, α ∈]0, π/2] and L > 0 be a real positive number.we denote by R (θ ,α) (L) the set of paths λ in • R originating from 0, piecewise C 1 , with length(λ ) < L + 1, with the conditions:• either λ stays in the open disc D(0, 1); • or else, for every t ∈ [0, 1], the right and left derivatives λ (t) do not vanish and arg λ (t) ∈]α + θ , θ + α[.

1 Fig. 4 .

 14 Fig. 4.11 Two paths belonging to R (θ ,α) (L) for θ = 0 and L ≥ 2.

Lemma 4 . 13 .

 413 The set R (θ ,α) (L) is an open and connected neighbourhood of 0 in R and R (θ ,α) (L) ⊂ R(0)

  3 and proposition 3.14). To each interval I j =]0, π[+ jπ, j ∈ Z, one associates the Borel-Laplace sum w tri, j (z) = S I j w (0,0) (z) ∈ O • D(Ij, τ) where • D(Ij, τ) is a sectorial neighbourhoods of ∞ with aperture(

. 15 ) 5 . 1 . 1 0-

 15511 Remark Let us consider the action of the gauge transform y = T 0 (z)u on the differential equation ∂ u +B 0 u = 0. This differential equation is transformed into the system ∂ y + A 0 y = 0 with A 0 = T 0 B 0 T -

. 33 ) 5 . 2 .

 3352 LemmaThe n-parameters formal expansion w(z, U) = ∑ k∈N n

i | = 1 Formula ( 5 .

 15 36) with k = e 1 provides D e 1 w e 1 = 0 (5.42)

15 4 .

 4 Symmetrically for k = e 2 , one gets λ 2 = -1, τ 2 = -3 2 as a necessary condition and D e 2 w e 2 = 0 (5.43)

Fig. 5 . 2 2 Lemma 5 . 7 .

 52257 Fig. 5.2 The Newton polygon at infinity N ∞ (D e 1 ) associated with the linear operator (5.45).

4 1z 2 +

 42 ∂ F(z, w 0 ) ∂ w w e 1 and we see that the right-hand side of this equation belongs to z -2 C[[z -1 ]] once w e 1 belongs to C[[z -1 ]], because of lemma 5.6 and to the choice of the coefficient τ 1 .

5. 4 . 1 . 2

 412 Resonances : first consequences Equation (3.6) has the feature to have positively resonant multipliers λ 1 = 1, λ 2 = -1 because λ λ λ .n = 0, for every n ∈ N (see definition 5.2). This brings semipositively resonances, the cases of semi-positive resonances being all described by λ 1 = λ λ λ .(n + e 1 ) and λ 2 = λ λ λ .(n + e 2 ), for every n ∈ N .

Definition 5 . 8 .

 58 We equip the graded algebra l∈N log l (z)C[[z -1 ]] with the valuation val defined by: val ∑ l log l (z) f l = min l {val f l }.

  z)C[[z-1 ]] and val w = val f . Moreover, if f = ∑ p l=0 log l (z) f l and w = ∑ p l=0 log l (z) w l , then:1. w p solves the ODE: P

1+e 1 (

 1

  fines a holomorphic function on • R (0) with at most exponential growth of order 1 at infinity. Moreover, w [0]

  have |Q k |(ξ ) = 3(|k| -1) ξ + λ λ λ .k (see (5.39)), Therefore, by lemma 6.1, |Q k |(|ζ |) |P k (-ζ )| ≤ 3M ρ,(0) . In the same way, one easily sees that |R k (∂ )| ≤ 9 4 |k|(|k| -1) (cf. (5.39)), thus the result by lemma 6.1. We eventually introduce the following notation that complements definition 3.10. Definition 6.1. Assume that G(ζ , w) = ∑ |l|≥0 c l (ζ )w l is an analytic function on the open polydisc ∆ r = ∏ n i=0 D(0, r i ). One defines the function |G|, analytic on ∆ r , by |G|(ξ , w) = ∑ l≥0 |c l |(ξ )w l .

Lemma 6 . 3 .

 63 The e i -st series w e i is 1-Gevrey. Its formal Borel transform reads B( w e i ) = δ + w e i and w e i is holomorphic on • R (0) with at most exponential growth of order 1 at infinity. More precisely, for every ρ ∈]0, 1[, there exist A > 0 and τ > 0such that for every ζ ∈ • R (0) ρ , | w e i (ζ )| ≤ Ae τ|ζ | .In the above upper bounds one can choose A = τ =5.81

  Equation (6.4) thus translates into a fixed point problem v e 1 = N( v e 1 ) where N : B r → B r is a contractive mapping for ν large enough. This ensures the existence and uniquess of w e 1 ∈ O( • R (0) ).

4 4 + 1 *

 41 +1 * w 0 (|ζ |) where w 0 (ξ ) = Ae τξ stands for the majorant function of w 0 given by theorem 3.3 and corollary 3.1, thus with A = 4.22 and τ = 4.22 ρ . Viewing the Grönwall-like lemma 3.9, one sees that for every ζ ∈ • R (0) ρ , | w e 1 (ζ )| ≤ w e 1 (|ζ |) where w e 1 solves the convolution equation: 1 M ρ,(0)w e 1 = 1 w 0 * (δ + w e 1 ).(6.5)

  lemma 4.11. We get that for every ζ ∈ • R (0) ρ | w Ke 1 (ζ )| ≤ w Ke 1 (ξ ) with ξ = |ζ |, where w Ke 1 is the entire function, real positive on R + , with at most exponential growth of order 1 at infinity, satisfying the linear equation: 1 M ρ,(0)

( 1 )

 1 ρ , | w Ke 1 (ζ )| ≤ w Ke 1 (ξ ) where ξ = leng(ζ ), with w Ke 1 an entire function, real positive and nondecreasing on R + , with at most exponential growth of order 1 at infinity, satisfying the linear equation: 1 M ρ,(1)

9 4ρ ξ + 4 + w * 2 .= 3 z 2 when 2 k.

 942322 lemma 6.3), and we recall that ∂ 2 F ∂ w 2 (ξ , w 0 ) = ξ while ∂ F ∂ w (ξ , w 0 ) = 4ξ + ξ * w 0 (ξ ). Therefore, w solves the convolution equation: ρ w = 3 + 4 + The generating function w(ξ,V) is thus the Borel transform of w(ζ,V), solution of the algebraic equation ρ w V → 0 with |z| large enough. We view (6.14) as a fixed point problem w = N(w). We set U = D(∞, 4ρ 27 ) × D(0, 2 3 √ ρ ), we equip the space O(U) with the maximum norm and we consider the closed ball B 1 = {w ∈ O(U), w ≤ 1} of the Banach algebra O(U), . One easily shows that N : B 1 → B 1 is a contractive map (remember that ρ < 1), hence the fixed-point problem w = N(w) has a unique solution w = w(z,V) in B 1 . Its Taylor expansion with respect to V at 0 reads w(z,V) = ∞ ∑ k=2 V k w ke 1 (z), where ( w ke 1 ) k≥2 is a sequence of holomorphic functions on the disc D(∞, 4ρ 27 ) and, by the Cauchy inequalities, for every integer k ≥ 2, sup |z|> 27 4ρ | w ke 1 (z)| ≤ 3 √ ρ Moreover, since

Theorem 6 . 2 .

 62 The transseries solutions of the prepared equation (3.6) associated with the first Painlevé equation,

Fig. 6 . 2

 62 Fig. 6.2 The (shaded) domain

Fig. 7 . 1 1

 71 Fig. 7.1 The Hankel contour γ [θ -2π,θ ],ε for θ = 0.

Definition 7 . 1 .

 71 Let I ⊂ S 1 be an open arc. For 0 ≤ r < R ≤ ∞, we denote by R r (I) the simply connected domain of C of the form R r (I) = {ζ = ξ e iθ | θ ∈ I, ξ ∈]r, R[}. One denotes by ¯R r (I) the closure of R r (I) in C We use abridged notations 0 (I), ¯0(I), ∞ (I) and ¯∞(I) for sectors, when R or r is unspecified. For any continuous function R : S 1 →]0, +∞[, we denote by R 0 the simply connected domain defined by R 0

Definition 7 . 2 . 2 0

 722 Let I ⊂ S 1 be an open arc. One says that two functions ϕ 1 ∈ O( (I)) define the same sectorial germ ∨ ϕ of direction I at 0, when ϕ 1 and ϕ 2 coincide on a same domain of type • 0 (I). We denote by O 0

Definition 7 . 5 .

 75 Let I ⊂ S 1 be any open arc of length ≤ π. One sets Ǒ0 (I) = O 0 ( Ǐ) and Ǒ0 (I) is called the space of germs of codirection I at 0. We denote by Ǒ0 the corresponding sheaf over S 1 .

Definition 7 . 6 .

 76 Let I ⊂ S 1 be any open arc of length ≤ π. We denote by ϕ = sing I 0 ∨ ϕ ∈ C (I) the microfunction of codirection I at 0 defined by the sectorial germ ∨ ϕ ∈ Ǒ0 (I) of codirection I.

Example 7 . 2 . 1 .

 721 For any n ∈ N, the sectorial germ ∨ I -n (ζ ) = (-1) n 2iπ n! ζ n+1 can be seen as a global section of the sheaf O 0 . The associated microfunction is equally denoted by I -n , δ (n) or by sing 0 ∨ I -n . Notice that for any holomorphic germ ϕ ∈ O 0 , the sectorial germ ϕ ∨ I 0 defines a microfunction sing 0 ( ϕ

7 .Lemma 7 . 1 .

 771 Let ε > 0 be a real psoitive number and I ⊂ S 1 be an open sector of length < π. We set S ε ( I) = η∈ • ∞ 0 ( I) D(η, ε), the "ε-neighbourhood" in C of the sector • ∞ 0 ( I). When the open arc I is of length = π, then I = {θ } and we set S ε ( I) = s∈R + D(se iθ , ε). We set • S ε (I) = C \ S ε ( I) and we denote by -∂ • S ε (I) = ∂ S ε ( I) the oriented boundary. We denote by Γ I,ε,η 1 ,η 2 the curve that follows the oriented boundary -∂ • S ε (I) from η 1 to η 2 . We denote by Γ I,ε the endless curve that follows the oriented boundary -∂ • S ε (I). Let ζ -S ε ( I) be the convex domain deduced from S ε ( I) by the point reflection centered on ζ /2 ∈ C. If dist(ζ , S ε ( I)) ≥ 2ε, then ζ -S ε ( I) ⊂ • S ε (I). In particular, for every ζ ∈ • S 2ε (I), for every η ∈ (-∂ • S ε (I)), one has ζη ∈ • S ε (I). Proof. We only consider the case where I ⊂ S 1 is an open arc of length < π. We pick an open sector • ∞ 0 ( I) and ζ ∈ C \ • ∞ 0 ( I). Then ζ /2 ∈ C \ • ∞ 0 ( I) as well. We denote by ζ -• ∞ 0 ( I) the convex domain deduced from • ∞ 0 ( I) by the point reflection centered on ζ /2 ∈ C. One sees that for every ξ ∈ ζ -• ∞ 0 ( I), for every η ∈ • ∞ 0

Fig. 7 . 2 .

 72 Fig. 7.2 The domain S ε ( I) (left-hand side shaded domain), the domain ζ -S ε ( I) (right-hand side shaded domain.

Fig. 7 . 3

 73 Fig. 7.3 Picture associated with the proof of lemma 7.2.

(

  Ǐ1 ) with R > r > 0 small enough. In what follows, we choose ε ∈]0, r 2 sin(π -| I|)[. We remark that both • S 2ε (I) ∩ D(0, r) and • S ε (I 1 ) ∩ D(0, R) are non empty domains and

  drawn on Fig. 7.4. For any η ∈ Γ I 1 ,ε,η 1 ,η 2 and any ζ ∈ • S 2ε (I) ∩ D(0, r), |ζ -η| < R+r and we know by lemma 7.1 that ζη ∈ • S ε (I). Therefore, the function

Fig. 7 . 4

 74 Fig. 7.4 The path of integration Γ I 1 ,ε,η 1 ,η 2 .

  Finally replacing ε by a another ε ∈]0, r 2 sin(π -| I|)[ yields the same conclusion : for ζ on the intersection domain • S 2ε (I) ∩ • S 2ε (I) ∩ D(0, r), one can compare the two functions

Definition 7 . 8 .

 78 Let be I an open arc I of length < π. We consider two microfunctions of codirection I, ϕ and ψ , represented by the sectorial germ of codirection I, ∨ ϕ and ∨ ψ respectively. For a covering of I by open arcs I 1 ⊂ I, the family of functions

Proposition 7 . 1 .

 71 glue together modulo O 0 and provide a microfunction of codirection I denoted by ϕ * ψ . It is called the convolution product of ϕ and ψ . The sheaf of microfunctions C is a sheaf of C-differential convolution algebras, for the derivation ∂ : sing I 0 ( ∨ ψ ) → sing I 0 (-ζ ∨ ψ ). These algebras are commutative, associative and with unit δ = sing In what follows we use the previous notations : ϕ and ψ are two microfunctions of codirection I, an open arc of length < π. One pick a subarc I 1 I and the microfunctions can be represented by functions ∨ ϕ and ∨ ψ belonging to O • ¯R+r 0 ( Ǐ1 ) with R > r > 0 small enough.

ψ

  defines a holomorphic function on the simply connected domain • S ε+ε (I 1 ) : just apply the Lebesgue dominated convergence theorem for ζ on any connected compact subset of • S ε+ε (I 1

ψ

  (equation(7.6)) for the convolution product ϕ * ψ of two microfunctions. Other representations can be obtained under convenient hypotheses as exemplified by the next proposition. Proposition 7.2. Let ψ be a microfunction of codirection I, an open arc of length < π, represented by the sectorial germ ∨ ψ of codirection I. Let be ϕ ∈ Γ (S 1 , C ) a microfunction of the form sing 0 ϕ log 2iπ with ϕ ∈ O 0 . Then, the microfunction ϕ * ψ of codirection I can be represented modulo O 0 by a family of functions of the form

  η

Fig. 7 . 5

 75 Fig. 7.5 Decomposition of the path Γ I 1 ,ε,η 1 ,η 2 .

Definition 7 . 11 .

 711 One defines the spaces SING ω , resp. SING ω,θ ,α of singularities at ω ∈ C, by translation from SING, resp. SING θ ,α . It is of course enough to study the spaces of singularities at 0 and this is what we do in what follows.

∨

  ϕ holomorphic in the domain 0 ( Ǐ) ⊂ C and such that lim ζ →0 ζ ∨ ϕ (ζ ) = 0 uniformaly in any

  ANA θ ,α that can be represented by holomorphic functions on a sector 0 ( Ǐ). By projection on C, one can think of ∨ ϕ , ∨ ψ as belonging to O( • 0 ( Ǐ)), that is sectorial germs of codirection I. By restriction, ϕ , ψ are seen as microfunctions of codirection I, whose convolution product ϕ * ψ ∈ Γ (I, C ) can be represented either by

Definition 7 . 19 .

 719 Let I ⊂ S 1 be an open arc and r ≥ 0 be a nonegative real number. we denote by: 1. A ≤0 ( • ∞ r (I)) the C-differential algebra of holomorphic functions ϕ on • ∞ r (I) that satisfy the property : for any proper subdomain • ¯∞ • ∞ r (I), for any ε > 0, there exists C > 0 so that for all z ∈ • ¯∞, |ϕ(z)| ≤ Ce ε|z| ; 2. A ≤0 (I) = lim -→ r→∞ A ≤0 ( • ∞ r (I)). This defines a presheaf A ≤0 ; 3. A ≤0 the sheaf over S 1 associated with the presheaf A ≤0 .

Lemma 7 . 3 .

 73 The space A ≤-1 ( • ∞ ), resp. A ≤-1 (I), of 1-exponentially flat functions on • ∞ , resp. of 1-exponentially flat germs at infinity over I, is a differential ideal of A ≤0 ( • ∞ (I)) -resp. of A ≤0 .

  backward, 2. then the circular arc δ J,ε = {εe iθ | θ ∈ J} oriented in the anti-clockwise way, 3. finally R θ 2 ,ε , resp. R θ 2 ,ε;κ , forward. Let us pick an open arc I of S 1 of length ≤ π, and a microfunction ϕ ∈ C (I) of codirection I, represented by the germ ∨ ϕ ∈ Ǒ0 (I). For any open arcI 1 =]α 1 , β 1 [ with Ī1 I, one can find R > 0 so that the restriction of ∨ ϕ to Ǐ1 =]α 1 -3π/2, β 1π/2[⊂ S 1is represented by a function (still denoted by ∨ ϕ ) holomorphic in the sector R 0 ( Ǐ1 ).

Fig. 7 . 6 2 θ•

 762 Fig. 7.6 Formal Laplace transform. The open arcs I 1 , I 2 , Ǐ1 , Ǐ2 , and the path γ = γ [θ 1 ,θ 2 ],ε;κ .

• Π θ 1 r 1 r 2 r

 112 contains any closed sector of the form• ¯∞ r (J 1 ) with J 1 an open arc so that J1 ⊂] -π 2θ 1 , -θ 1 + π 2 [ and r > 0 large enough. Since β 2 -π 2 < θ 1 < α 2 + π 2 ,one deduces that • Π θ contains any closed sector of the form • ¯∞ r (I 2 ) with r > 0 large enough. Similarly, contains any closed sector of the form • ¯∞ r (I 2 ) with r > 0 large enough.

Example 7 . 3 .∨

 73 For (σ , m) ∈ C × N and I =]π/2, π/2[∈ S 1 , we consider the microfunction Jσ,m= sing I 0 Jσ,m ∈ C (I) represented by the sectorial germ ∨ Jσ,m= ∂ ∂ σ ∨ I σ ∈ Ǒ0 (I) and the branch of the log such that arg(log ζ ) ∈ Ǐ =] -2π, 0[. By standard formulae recalled in Sect. 7.1, one readily gets that its formal Laplace transform Jσ,m= L (I) Jσ,m is an asymptotic class that can be represented by the (sectorial germ at infinity of) holomorphic function(s) (-1) m log m (z) z σ ∈ A ≤0 (I ), I =]π/2, π/2[ with the determination of the log so that arg(log z) ∈ I . The following proposition is a straight consequence of the very construction of the formal Laplace transform. Proposition 7.7. The formal Laplace transform L : C → A ≤0 /A ≤-1 satisfies the identity : L • ∂ = ∂ • L .

7. 5 . 2 Formal

 52 Borel transform for asymptotic classesLet I ⊂ S 1 be an open arc with length ≤ π and ϕ ∈ A ≤0 (I ) be a sectorial germ at infinity. For any open arc I 1 I , one can find r > 0 so that the restriction of ϕ to I 1 is (represented by) a holomorphic function (still denoted by ϕ) on the domain• ∞ r (I 1 ). We set ∨ ϕ z 1 ,α (ζ ) = -1 2iπ R α,z 1e zζ ϕ(z)dz for any z 1 ∈• ∞ r (I 1 ) and any direction α ∈ I 1 , see Fig.7.7. We can make the following observations about this Laplace integral ∨ ϕ z 1 ,α :

  Fig. 7.7 Formal Borel transform. The open arcs I , and the path R α,z 1 .

Remark 7 . 3 .

 73 whose compatibility with the restriction maps is easy to check. Definition 7.22. The morphism of sheaves B :A ≤0 /A ≤-1 → C is called the formal Borel transformThe formal Laplace transform for microfunctions and the formal Borel transform for asymptotic classes are isomorphisms of sheaves, as shown in[1] to whom we refer:Proposition 7.8. The morphisms L : C → A ≤0 /A ≤-1 and B : A ≤0 /A ≤-1 → C are isomorphisms of sheaves and L • B = Id, B • L = Id. The morphism of sheaves ϕ ∈ O 0 → ϕ = sing I 0 ϕ log 2iπ ∈ C (I)is injective as already mentioned. The following commutative diagram makes a link between the formal Laplace transform for regular minor, resp. formal Borel transform for 1-Gevrey formal series, and the formal Laplace transform for microfunctions, resp. formal Borel transform for asymptotic classes:

Definition 7 . 25 .

 725 ζ )dζ , with ∨ ϕ any major of ϕ . This introduces the following definition. (Notice that ( J = J ). The morphism L β = L (

r 1 >(ϕ 1 ϕ 2 ) z 1 ,α 1

 1211 r, and for any direction α 1 ∈ ( β . The function ∨

  nothing but a major of the singularity sing 0e z 1 ζ 2iπζ * ϕ 1 * ϕ 2 .But sing 0 e z 1 ζ 2iπζ = δ and therefore sing 0∨ (ϕ 1 ϕ 2 ) z 1 ,α 1 = ϕ 1 * ϕ 2 .From Proposition 7.8, we know that B • L = Id (when considering B and L as morphisms of sheaves), thus the conclusion. The last statement as been already seen.

7. 6 . 1

 61 Laplace transforms Definition 7.29. Let I ⊂ S 1 be an open arc and r ≥ 0. We denote by: 1. E ≤1 (I) the C-differential algebra of holomorphic functions ϕ on • ∞ 0 (I) with 1-exponential growth at infinity on the direction I : for any proper subsector • ¯∞ • ∞ 0 (I), there exist C > 0 and τ > 0 so that, for all z ∈ • ¯∞, |ϕ(z)| ≤ Ce τ|z| ; 2. when I is of length ≤ π, Ě ≤1 (I) = E ≤1 ( Ǐ) is the space of holomorphic functions ϕ on • ∞ 0 ( Ǐ), with 1-exponential growth at infinity on the codirection I. 3. E ≤1 , resp. Ě ≤1 , the sheaf over S 1 corresponding to the family E ≤1 (I) , resp. Ě ≤1 (I); 4. O(C) ≤1 the space of entire functions with 1-exponential growth at infinity on every direction. Pick an open arc I ⊂ S 1 of length ≤ π, and a function ∨ ϕ ∈ Ě ≤1 (I). Thus ∨ ϕ is holomorphic on • ∞ 0 ( Ǐ) and for any open arc I 1 so that Ī1 ⊂ I, for any ε > 0, there exist C > 0 and τ > 0 so that, for all ζ ∈ • ¯∞ ε ( Ǐ1 ), | ∨ ϕ (ζ )| ≤ Ce τ|ζ | . We consider the following Laplace integral,

R θ 1

 1

  1 ), once r > 0 is chosen large enough.Therefore, ϕ I 1 belongs to the space A ≤0 ( • ∞ r 1 (I 1 )) for r 1 > 0 large enough, because ε > 0 can be chosen arbitrarily small. It is easy to see that adding to ∨ ϕ any element of O(C) ≤1 , does not affect the function ϕ I 1 (just deform the contour of integration, by Cauchy).The family of functions (ϕ I 1 ) I 1 ⊂I obtained this way glue together analytically, by Cauchy. The above construction gives a morphism, L (I) : Ě ≤1 (I)/O(C) ≤1 → A ≤0 (I ), compatible with the restriction maps, which provides a morphism of sheaves 3 . Definition 7.30. The morphism of sheaves L : Ě ≤1 /O(C) ≤1 → A ≤0 is called the strict Laplace transform 4 .

7. 5 . 2 .

 52 We pick an open arc I ⊂ S 1 of length ≤ π and ϕ ∈ A ≤0 (I ). For z 1 ∈ • ∞ r (I ), r > 0 large enough, for any direction α ∈ I , we set

Proposition 7 . 11 .

 711 The morphisms of sheaves L : Ě ≤1 /O(C) ≤1 → A ≤0 and B : A ≤0 → Ě ≤1 /O(C) ≤1 are isomorphisms of sheaves of C-differential algebras, and L • B = Id, B • L = Id. 7.6.2 Singularities and Laplace transform 7.6.2.1 Summable singularities

σ 2 .

 2 Thus by proposition 7.10, L I σ 1 * I σ 2 = 1 z σ 1 +σ 2 and one concludes by formal Borel transform. Same proof for the other equality.

Proposition 7 . 14 .

 714 The space SING sum θ ,α is a commutative and associative algebra with unit δ . The Laplace transformL β : SING sum θ ,α → A ≤0 ( ( β ) is compatiblewith the convolution of singularities:L β ϕ * ψ = L β ϕ L β ψ . Moreover, L β ( ∂ ϕ ) = ∂ L β ϕ .Proof. We go back to the very definition of the convolution product of microfunctions and singularities. For ϕ , ψ ∈ SING θ ,α , for any β ∈ J =]α + θ , θ + α[, the convolution product ϕ * ψ can be represented, for ζ ∈ • S 2ε (]β -2π, β [) with ε > 0 as small as we want, by

  β -2π, β [))

∨ϕ

  with the following properties:1. the restriction of ∨ ϕ as a sectorial germ of codirection I =]π/2 + θ , θ + π/2[, can be represented by a function Φ holomorphic on the cut planeC \ [0, e iθ ∞[= • ∞ 0 ( Ǐ), Ǐ =] -2π + θ , θ [; 2. Φ is bounded on • S ε (I), for every ε > ε.3. Φ can be analytically continued to C \ Z.Proof. Let∨ ϕ 1 be a major of ϕ which can be analytically continued to C \ Z. This major can be represented by a function Φ 1 holomorphic on • R 0 ( Ǐ) ∪ S 2ε ( I) \ [0, e iθ ∞[, for R > 0 and ε > 0 small enough, and Φ 1 can be analytically continued to C \ Z. The boundary Γ I,ε = -∂ • S ε (I) can be seen as an embedded curve H 0 : R → C that fulfills the condition of lemma 7.5 : one can find a function Ψ 1

  are equal to the identity in a neighbourhood of infinity, Fig. 7.8. Finally, from the fact that Φ = Φ 1 +Ψ with Ψ 1 +Ψ 2 ∈ O 0 , we see that Φ defines a sectorial germ ∨ ϕ of codirection I =]π/2 + θ , θ + π/2[ whose associated microfunction coincides with the restriction of ϕ to the codirection I.

Fig. 7 . 8

 78 Fig. 7.8 Deformation of the contour Γ I,ε by an isotopy equal to the identity in a neighbourhood of infinity, for θ = 0.

Definition 7 . 33 .

 733 Let θ ∈ S 1 be a direction. We set• R θ ,(0) = C \ [0, e iθ ∞[. Let ζ 0 be a complex number in • R θ ,(0) \ Z. We denote by A θ ,ζ 0 (resp. B ζ 0 ) the set of paths in • R θ ,(0) (resp. C \ Z) originating from ζ 0 ,endowed with the equivalence relation ∼ • R θ ,(0) (resp. ∼ C\Z ) of homotopy of paths with fixed extremities. We set R θ ,ζ 0 = A θ ,ζ 0 ∪ B ζ 0 and we denote by (θ ,ζ 0 ) ∼ the relation on R θ ,ζ 0 defined as follows. For any two γ 1

  14) defines a holomorphic function on• S 2ε (I) ⊂ • R θ ,(0) which by(7.8), represents the convolution product ϕ * ψ . We want to show that χ can be analytically continued onto the Riemann surface (R θ , p) (thus to C \ Z as well). We choose a pointζ 0 ∈ • S 2ε (I) so that {ζ 0 -H 0 } ∩ Z = /0, and we view χ as a germ of holomorphic functions at ζ 0 : for ξ ∈ C close to 0, χ(ζ 0

7. 8 . 1 1 0 2 0

 8112 Alien operators associated with a triple 7.8.1.1 Mains definitions We consider two directions θ 1 , θ 2 ∈ S 1 , a point ω ∈ Z and a sectorial germ ∨ ϕ ∈ O 0 θ 1 of direction θ 1 . We can think of ∨ ϕ as a sectorial germ on a sector • R 1 0 (I 1 ) for 0 < R 1 < 1 and I 1 ⊂ S 1 an open arc bisected by θ 1 , and this is what we do in what follows.We now assume that ∨ ϕ can be analytically continued to C \ Z. We consider a path γ :J → C \ Z starting from ζ 1 ∈ • R (I 1 ) and ending at ζ 2 close to ω so that ζ 2ω ∈ • R (I 2 ) with 0 < R 2 <1 and I 2 ⊂ S 1 an open arc bisected by θ 2 . See Fig. 7.9. By hypotheses, the analytic continuation (cont γ ∨ ϕ ) of ∨ ϕ along γ is a well-defined germ of holomorphic functions at ζ 2 that only depends on the homotopy class of γ (for the relation of homotopy of paths in C \ Z with fixed extremities). Moreover, if ∨ ψ ∈ O ζ 2 -ω stands for the germ of holomorphic functions at ζ 2ω defined by ∨ ψ (ξ ) = cont γ ∨ ϕ (ω + ξ ) then, still by analytic continuations, ∨ ψ determines a unique sectorial germ on • R 2 0 (I 2 ) and thus, by restriction, a unique sectorial germ ∨ ψ ∈ O 0 θ 2 . This justifies the following definition adapted from [18]. Definition 7.44. Let be θ 1 , θ 2 ∈ S 1 , ω ∈ Z and ∨ ϕ ∈ O 0 θ 1 a sectorial germ of direction θ 1 that can be analytically continued to C \ Z. Let γ : J → C \ Z be a path starting from a sufficiently small sector • 0 (I 1 ) bisected by θ 1 and ending close to ω in a sufficiently small sector of the form ω + • 0 (I 2 ) where I 2 bisects θ 2 . Then, one denotes by •

1 Fig. 7 .

 17 Fig. 7.9 A triple (γ, θ 1 , θ 2 ) defining the operator A γ ω (θ 1 , θ 2 ) at ω = -2.

θ 1 =A

 1 π(θ 1 ) ∈ S 1 which can be analytically continued to C \ Z. Therefore, under the conditions of definition 7.The alien operators have their counterparts on asymptotic classes through formal Borel and Laplace transforms. Definition 7.46. The alien operator A γ ω (θ 2 , θ 1 ) at ω associated with the triple (γ, θ 1 , θ 2 ) is defined on asymptotic classes by making the following diagram commuting: RES

Lemma 7 . 8 .

 78 Let be θ 1 ∈ {πk, k ∈ Z} ⊂ S 1 . For any integer k ≥ 1, any ε ∈ {-, +} and any m 1 ∈ N , setting ω 1 = e i • θ 1 , the alien operator A

For

  any singularity of the form ϕ = sing 0 ∨ ϕ ∈ SING, ∨ ϕ ∈ ANA, we write ρ k . ϕ = sing 0 ( ∨ ϕ • ρ k ) ∈ SING. More generally, for any r ∈ R, one sets ρ r : ζ = re iθ ∈ C → ρ r (ζ ) = re iθ +2iπr ∈ C and ρ r . ϕ = sing 0 ( ∨ ϕ • ρ r ) ∈ SING. Remark 7.8. With this notation, the variation map var : SING → ANA reads var = Idρ -1 .

Fig. 7 .Proposition 7 . 19 .

 7719 Fig. 7.10 Two triples (γ, θ 1 , θ 2 ) and (γλ ω , θ 1 , θ 2 ) for the point ω = -2, with λ ω a closed path of winding number wind ω (λ ω ) = 1 at ω.

Proposition 7 . 21 .

 721 For any alien operator of the form A γ ω (θ , θ ) acting on RES Z or RES ( • θ ,α) (L), for any singularity ϕ and any resurgent constant const∈ CONS,A γ ω (θ , θ ) const * ϕ = const * A γ ω (θ , θ ) ϕ . (7.22)We stress that in proposition 7.21, only alien operators of the form A γ ω (θ 2 , θ 1 ) with θ 1 = θ 2 are considered. We omit the proof of this proposition which relies on a careful reading of what have been done for showing theorem 7.1.

•

  θ m given by(7.23), and θ m ∈ S 1 so that• θ m = π(θ m ). Then the alien operator A γ ω m (θ m , θ 1 )at ω m associated with the triple (γ, θ 1 , θ m ) can be written as a Z-linear combination of composite operators of the form A

  γ

  -, +, +, +, -), n = (1, 2, 1, 1, 1) and take θ 1 = 0, see Fig.4.3. Using (7.23), we define:

Fig. 7 .

 7 Fig. 7.13 A path of type γ

7 . 27 )

 727 where the sum runs over allω 1 = m 1 e iθ , ω 2 = m 2 e iθ , with m 1 , m 2 ∈ N such that m 1 + m 2 = m. The alien operators ∆ ω : RES ( • θ ,α) (L) → SING θ ,α , resp. ∆ ω : RES → RES, satisfy the Leibniz rule, ∆ ω ϕ * ψ = ∆ ω ϕ * ψ + ϕ * ∆ ω ψ . Moreover, ∆ ω ( ∂ ϕ ) = ( ∂ -• ω)(∆ ω ϕ ). Eventually, ∆ +ω cons= ∆ ω cons= 0 for any resurgent constant cons.

Theorem 7 . 4 .

 74 For any θ ∈ { kπ, k ∈ Z}, ω ∈ C with arg(ω) = θ , ∆ ω = ∑ s∈N (-1) s-1 s ∑ arg(ω 1 )=•••=arg(ω s-1 )=θ 0≺

Fig. 7 .Theorem 7 . 5 .

 775 Fig. 7.14 Symmetrically contractile path H 1 and contributions to ∆ + ω ϕ * ψ for ω = 3. Pinchings occur between 1 and ζ -2, and between 2 and ζ -1.

• θ 1 ε

 1 n for ε = (+, -, +), n = (1, 3, 1) and we have shown the identity: 0) = ∆ + 4 -∆ + 2 • ∆ + 2ρ -1 .∆ + 2e -2iπ • ∆ + 2 .

Example 7 . 15 .

 715 We consider ϕ(ζ ) = ζ e 2iπζ -1 ∈ R. This is a meromorphic function with simple poles at Z whose residue at m ∈ Z is res m ϕ = m. Introducing the singularity ϕ = ϕ, one easily deduces that for every k ∈ Z and every m ∈ N ,∆ me iπk ϕ = ∆ + me iπk ϕ = (-1) k mδ . (7.30)The formal Laplace transform L ϕ is an asymptotic class ϕ = ϕ that can be representend by a Z-resurgent series ϕ ∈ R Z and (7.30) translates into∆ me iπk ϕ = ∆ + me iπk ϕ = (-1) k m.(7.31)We now look at the singularity ψ σ ,n = Jσ,n * ϕ for (σ , n) ∈ C × N. By the Leibniz rule and since Jσ,n is a resurgent constant,∆ me iπk ψ σ ,n = Jσ,n * ∆ me iπk ϕ = (-1) k m Jσ,n ∈ α>0 SING πk,α .(7.32)The asymptotic class associated to ψ σ ,n by formal Laplace transform isψ σ ,n = Jσ,n ϕ ∈ RES.The identity (7.32) provides: ∆ me iπk ψ σ ,n = (-1) k m Jσ,n ∈ α>0 ASYMP πk,α . (7.33) 7.8.3.3 The spaces RES (k)

Corollary 7 . 2 .

 72 Let be k ∈ N and ω ∈ C such that • ω is an integer and |ω| ≤ k. The alien operator ∆ ω acts on RES(k) and∆ ω : RES (k) → RES (k-|ω|) , when 1 ≤ |ω| ≤ k -1 ∆ ω : RES (k) → SING arg(ω),π , when |ω| = k. (7.34)Moreover for any ϕ , ψ ∈ RES (k) :• ∆ ω ( ∂ ϕ ) = ( ∂ -• ω)(∆ ω ϕ );• ∆ ω ϕ * ψ belongs to RES(1) when 1 ≤ |ω| ≤ k -1 and to SING arg(ω),π when| • ω| = k and furthermore ∆ ω ϕ * ψ = ∆ ω ϕ * ψ + ϕ * ∆ ω ψ (Leibniz rule).Proof. The identity (7.34) is a consequence of proposition 7.23. The commutation formula [∆ ω , ∂ ] = -• ω ∆ ω ensues from proposition 7.18. Notice now that for any k ∈ N , any L ∈]k -1, k] and any α ∈]0, π/2], one has RES ( • θ ,α) (L) ⊃ RES (k) . Pick two singularities ϕ , ψ ∈ RES (k) and consider them as belonging to RES ( • θ ,α) (L).

Definition 7 .

 7 53. The alien operators ∆+ ω , ∆ ω : RES (k) → RES (k-|ω|) for 1 ≤ |ω| ≤ k -1, resp. ∆ + ω , ∆ ω : RES (k) → ASYMP arg(ω),π, for |ω| = k, for asymptotic classes are defined by making the following diagrams commuting:RES (k) ∆ + ω ,∆ ω -→ RES (k-|ω|) L ↓↑ B L ↓↑ B RES (k) ∆ + ω ,∆ ω -→ RES (k-|ω|) ., resp.RES (k) ∆ + ω ,∆ ω -→ SING arg(ω),π L ↓↑ B L ↓↑ B RES (k) ∆ + ω ,∆ ω-→ ASYMP arg(ω),π .

Definition 7 . 1 ↓↑ T 1

 711 54. A Z-resurgent singularity ϕ ∈ RES is said to be a simple resurgent singularity when ϕ = aδ + ϕ ∈ SING simp and, for any alien operator A γ ω (θ 2 , θ 1 ), A γ ω (θ 2 , θ 1 ) ϕ belongs to SING simp . The minor ϕ, resp. the 1-Gevrey series ϕ = a + L ϕ, associated with a simple Z-resurgent singularity is a simple resurgent function, resp. a simple resurgent series, and one denotes by RZ simp , resp. R Z simp the space of simple Z-resurgent functions, resp. series. The resurgent subalgebra made of simple resurgent singularities is denoted by RES simp Z and the corresponding space of asymptotic classes is denoted by RES simp Z . As usual in this course, we use abridged notations. One can make acting the alien operators on the space R simp . Definition 7.55. The alien operators ∆ + ω , ∆ ω : R simp → R simp are defined by making the following diagrams commuting: ↓↑R simp ∆ + ω ,∆ ω -→ R simp .Obviously (from proposition 7.26), for any ϕ ∈ R simp , the alien derivative ∆ ω ϕ only depends on• ω, thus one could define ∆ + ω , ∆ ω : R simp → R simp for ω ∈ Z .

(

  uniquely) determined by ϕ, ρ. ϕ, • • • , ρ d . ϕ. Let be ϕ ∈ Nils with d( ϕ) = d + 1.

Definition 7 .

 7 60. A Z-resurgent singularity ϕ ∈ RES Z is a ramified resurgent singularity when ϕ ∈ SING ram whereas, any alien operator A γ ω (θ 2 , θ 1 ), A γ ω (θ 2 , θ 1 ) ϕ belongs to SING ram . The space of ramified resurgent singularities makes a resurgent subalgebra denoted by RES ram Z . The corresponding space of asymptotic classes, resp. formal expansions, is denoted by RES ram Z , resp. R Z ram .

Proposition 7 . 27 .

 727 The formal expansion ϕ = ∑ (σ ,m) J σ ,m ϕ (σ ,m) ∈ Nils belongs to R ram if and only if each of its components ϕ (σ ,m) belongs to R ram . Definition 7.61. The alien operators ∆ + ω , ∆ ω : R ram → R ram are defined by making the following diagrams commuting: down a direct consequence of proposition 7.19. (We warn to the change of sign). Proposition 7.28. Let ϕ be an element of R ram . Then, for any ω ∈ C with • ω ∈ Z , for any k ∈ Z,∆ ωe 2iπk ϕ = ρ k . ∆ ω ρ -k . ϕ , ∆ ωe iπ ϕ = ρ 1/2 . ∆ ω ρ -1/2 . ϕ . Example 7.16. Suppose that ϕ ∈ C[[z -1 ]] 1 belongs to R ram with ∆ ω ϕ = log(z) z σ ψ, ψ ∈ C[[z -1 ]]. For k ∈ Z, ρ -k . ϕ(z) = ϕ(z), then ∆ ωe 2iπk ϕ(z) = log(z+2πk)z σ e 2iπkσ ψ(z).

Chapter 8 Resurgent 8 . 1

 881 structure for the firstPainlevé equation Abstract We show the resurgence property for the formal series solution of the prepared form associated with the first Painlevé equation. The detailed resurgent structure is given in Sect. 8.1. Its proof is given using the so-called bridge equation (Sect. 8.4), after some preliminaries (Sect. 8.3). The nonlinear Stokes phenomena are briefly analyzed in Sect. 8.2. The main theorem8.1.1 ReminderThe formal integral of the prepared ODE (3.6) associated with the first Painlevé equation was described with theorem 5.1 and its corollary 5.1. It can be written under the following equivalent form:w(z, U) = W 0 (z) + ∞ ∑ n=0 ∑ k∈Ξ n+1,0 \Ξ n,0

. 4 )

 4 P k = P k ( w 0 ) = P(-λ λ λ .k + ∂ ) + 1 z Q(-λ λ λ .k + ∂ ) -∂ F(z, w 0 ) ∂ w .

Proposition 8 . 2 . 1 )

 821 In theorem 8.1, the Stokes coefficients are A 0 (ω 0

0 .

 0 1 and f 2 obviously belong to CONS. Equation(8.3) translates into the fact that w 0 satisfies the following convolution equation:P( ∂ ) w 0 + I 1 * Q( ∂ ) w 0 = F (ζ , w 0 ) (8.13) = f 0 + f 1 * w 0 + f 2 * w * 2One can rather introduce the asymptotic class w 0 = w 0 ∈ ASYMP simp (cf. definition 7.27) and equation (8.3) becomes:P(∂ ) w 0 + 1 z Q(∂ ) w 0 = F(z, w 0 ) (8.14) = f 0 + f 1 w 0 + f 2 w 2 0

1 , 1 )= f 1

 111 rule and the commutation rule [∆ ω 0 , one deduces from (8.13) that W solves in SING 0,π the following associated linear convolution equation:P( ∂ -1) W + I 1 * Q( ∂ -1) W = ∂ F (ζ , w 0 ) +2 f 2 * w 0 * W .

3 2 3 2 2 W W e 1 ∂ W ∂ W e 1 ,C 1 = z -3 e -2z 2 W e 2z W e 2 ∂

 33211122 ) through the Taylor map, has its general formal solution that belongs to the direct sum Nils 1 ⊕ e 2z Nils 1 , under the formW (z) = C 1 z w e 1 (z) +C 2 e 2z z w e 2 (z) (8.18) = C 1 W e 1 (z) +C 2 e 2z W e 2 (z), where W e 1 and W e 2 belong to the space Nils 1 of 1-Gevrey Nilsson series. One should precise what we mean by "general formal solution". The linear operator P e 1 is of order 2 in z and the particular solutions W e 1 and e 2z W e 2 are two independent formal solutions : their wronskian is W e 1 e 2z W e 2 ∂ W e 1 ∂ (e 2z W e 2 ) = 2z 3 e 2z . Thus, if W belongs to a differential algebra that contains Nils 1 ⊕ e 2z Nils 1 as sub-vector space, for instance the direct sum ∏ k∈Z e -kz Nils 1 and if P e 1 ( w 0 ) W = 0, then W is of the form (8.18) with C 1 ,C 2 ∈ C given by the Kramer's formulas: C 2 = -z -3 e -2z W ∂ (e 2z W e 2 ) .

Lemma 8 . 1 . 1 w 0 ∈ω 0 1 w 0 = A 0 (ω 0 1 ) I - 3 2 * w e 1 = A 0 (ω 0 1 ) W e 1 ,ω 0 1 w 0 = A 0 (ω 0 1 ) W e 1 ∈ 1 w 0 = A 0 (ω 0 1 ) W e 1 ∈

 8110101311110111011 ) in ∏ k∈Z e -kz ASYMP 0,π is a linear combination of W e 1 ∈ ASYMP ram and e 2z W e 2 ∈ e 2z ASYMP ram with W e i = ram W e i . Consequently: There exists A 0 (ω 0 1 ) ∈ C such that the singularity ∆ ω 0 SING 0,π is of the form ∆ thus can be extended uniquely to an element of SING. In other equivalent words, ∆ ASYMP ram , ∆ ω 0 Nils 1 .

Lemma 8 . 2 . 2 . 1 n! w n is a 1 -

 82211 Let w ∈ z -1 C[[z-1 ]] 1 be a 1-Gevrey series with vanishing constant term, and w ∈ O 0 its minor. The following properties are satisfied.1. The formal series(1 + w) ∈ C[[z -1 ]] 1 is invertible. Its inverse (1 + w) -1 is 1-Gevrey and has a formal Borel transform B(1+ w) -1 ∈ Cδ ⊕ O 0 of the form (δ + w) * -1 = δ + ∑ n≥1 (-1) n w * n . The formal series log(1 + w) = ∑ n≥1 (-1) n+1 nw n is a 1-Gevrey with vanishing constant term, whose minor is given by log * (δ + w) := ∑ n≥1 (-1) n+1 n w * n . 3. The formal series w is exponentiable in the sense that its exponential e w = ∑ n≥1 Gevrey series, whose minor is of the form exp * ( w) := δ + ∑ n≥1 1 n! w * n . Moreover, log • exp = exp • log = Id.

( ∂ 2 -- 1 2 * w e 1 + I - 3 2* 1 .

 21131 ∂ ) S * W e 1 +2 ∂ S * ∂ W e 1 -3 I 1 * ∂ S * W e 1 = 0. (8.20) Since ∂ W e 1 = 3 2 I ∂ w e 1 , equation(8.20) reduces to the equation∂ 2 S= δ -2 χ * ∂ S, χ = w * -1 e 1 * ∂ w e 1 ,(8.21)where χ = χ is the singularity associated with the minor χ(ζ ) ofχ(z) = ∂ w e 1 w e 1 ∈ z -2 C[[z -1 ]]The formal series χ has a unique primitive χ 0 (z) = ∂ -1 χ(z) = log w e 1 (z) in the maximal ideal z -1 C[[z -1 ]] 1 of C[[z -1 ]]1 and, thus, χ 0 as well as its associated singularity χ 0 is exponentiable in SING. (Lemma 8.2) More simply, exp * ( χ 0 ) = δ + w e 1 , thus exp * (2 χ 0 ) = δ + (2 w e 1 + w * 2 e 1 ).

  What we have shown amounts to the following upshot. The solutions of the equation P e 1 ( w 0 ) W = 0 in the differential algebra ∏ k∈Z e -kz Nils 1 are spanned by the independent solutions W e 1 ∈ Nils 1 and e 2z W e 2 ∈ e 2z Nils 1 . This implies that the solutions of the equation P e 1 ( w 0 ) W = 0 in the differential algebra ∏ k∈Z e -kz ASYMP, resp.

Lemma 8 . 3 .

 83 For k ∈ N 2 , we denote by W k ∈ ASYMP ram the asympotic class defined by W k = ram W k where W k ∈ Nils 1 satisfies (8.4). Let θ ∈ S 1 be any direction, α > 0 and k ∈ N 2 \{0}. If W ∈ ∏ ∈Z e -z ASYMP θ ,α solves the linear differential equation

  A 0 (ω j i ) W e i , resp. ∆ ω j i w 0 = A 0 (ω j i ) W e i ,for any i = 1, 2 and j ∈ Z. Since W e 1 and W e 2 belong to RES

∆ 2ω j 1 w 0 ∈2ω j 2 w 0 ∈∆ 2ω j 2 w 0 ∈Lemma 8 . 4 . 1 w 0 .

 1020208410 ASYMP 2π j,π and ∆ SING 2π( j+1/2),π , resp. ASYMP 2π( j+1/2),π , are well-defined. As a matter of fact, these alien derivatives are quite simple ! For any ω ∈ C so that• ω = ±2, one has ∆ ω w 0 = 0. Equivalently,∆ ω w 0 = 0, ∆ ω w 0 = 0.Proof. We only calculate W = ∆ 2ω j Through the alien derivation ∆ 2ω j 1 , equation (8.3) is transformed into the linear ODEP(∂ -2) W + 1 z Q(∂ -2) W = ∂ F(z, w 0 ) ∂ w W (8.26)as a consequence of corollary 7.2. We recognize the equation P 2e 1 ( w 0 ) W = 0. By lemma 5.4, the general formal solution for the linear equationP 2e 1 ( w 0 ) W = 0 is of the form C 1 e z W e1 +C 2 e 3z W e 2 and we either conclude with the reasoning made in Sect. 8.3.2 (still write W under the form W = S * W e 1 and show that S= 0) or rather directly with lemma 8.3 : the solutions of the equation P 2e 1 ( w 0 ) W = 0 in ∏ k∈Z e -kz ASYMP are C 1 e z W e 1 +C 2 e 3z W e 2 and one concludes that ∆ 2ω j 1 w 0 = 0 since the alien derivative belongs to ASYMP 2π j,π .

• ω 1 =ω 2 =

 12 ±1 and • ±1. From what we know, this amount to showing that the alien derivatives ∆ ω 2 W e i belong to RES(1)

e 1 ∂ 2 F 1 W e 1 =

 1211 (z, w 0 ) ∂ w 2 , deduced from (8.27) through the Taylor map, reads:W = 2A 0 (ω 0 1 ) W 2e 1 +C 1 e z W e 1 +C 2 e 3z W e 2 ∈ ∏ k∈Z e -kz Nils 1 with C 1 ,C 2 ∈ C. By lemma 8.3 one gets ∆ ω 0 2A 0 (ω 0 1 ) W 2e 1 , which thus belongs to RES (1)by proposition 8.1. Of course, one can keep on that way, by induction. However, a lesson has to be learned from what precedes : the resurgent structure is closely coupled to the formal integral and it is much time to introduce the bridge equation.

8. 4 2 k 2 = 2 U

 4222 The bridge equation and proof of the main theoremWe go back to the formal integralw(z, U) = ∑ k∈N 2 U k e -λ λ λ .kz W k ∈ ∏ k∈Z e -kz Nils 1 [[U]](8.28)and we consider its derivatives with respect to the indeterminateU i , i = 1, 2: .e i U k-e i e -λ .kz W k ∈ ∏ k∈Z e -kz Nils 1 [[U]] (8.29) = W e i + O(U 1 ,U 2 ).Since the formal integral w solves the differential equationP(∂ ) w + 1 z Q(∂ ) w = F(z, w), one deduces that the following identity holds for i = 1, 2:P(∂ ) + 1 z Q(∂ ) -∂ F(z, w) ∂ w ∂ w ∂U i = 0, i.e. P 0 ( w) ∂ w ∂U i = 0. (8.30)The formal solutions for the equation P 0 ( w 0 ) W = 0 is spanned by e -λ 1 z W e 1 and e -λ 2 z W e 2 . Therefore, ∂ w ∂U 1 and ∂ w ∂U 2 are two linearly independent solutions for the order two linear differential equation P 0 ( w) W = 0 , explicitly (wronsk stands for the wronskianwronsk e -λ 1 z W e 1 , e -λ 2 z W e 2 = 2z 3 .Lemma 8.3 translates into the fact that for any series of the formW (z, U) = ∑ k∈N k W k , W k ∈ ∏ k∈Z e -kz ASYMP θ ,α , which satisfies the second order equation P 0 ( w) W = 0, there exist uniquely determined constants A(ω, U) ∈ C[[U]] and B(ω, U) ∈ C[[U]] such that W (z, U) = A(ω, U) ∂ w ∂U 1 + B(ω, U) ∂ w ∂U 2 , ∂ w ∂U i = ram ∂ w ∂U i (8.31)

•ω=

  ±1. By proposition 8.1 and corollary 7.2, the alien derivation ∆ ω acts on the formal integral w (z, U). As a matter of fact, it will be easier to use the dotted alien derivation,• ∆ ω = e -ωz ∆ ω which has the virtue of commuting with the derivation ∂ . Therefore,• ∆ ω w (z, U) = ∑ k∈N 2 U k e -λ λ λ .kz • ∆ ω W k , • ∆ ω W k ∈ e -ωz ASYMP arg(ω),πandP 0 ( w) • ∆ ω w = 0.We deduce that the decomposition (8.31) holds for• ∆ ω w. This decomposition • ∆ ω w= A(ω, U) ∂ w ∂U 1 + B(ω, U) ∂ w ∂U 2is the so-called bridge equation of Ecalle, that is a link between alien derivatives and the usual partial derivatives. Let Ξ ⊂ N 2 be the set defined by Ξ = Ξ 0 = {ke 1 , ke 2 | k ∈ N} and set Ξ n = n + Ξ for any n ∈ N . With these notations, the formal integral can be written as follows: -λ λ λ .kz W k (z) = e -λ λ λ .kz W k+n (z)(8.33)

  38. H. Umemura, Galois theory and Painlevé equations. In Théories asymptotiques et équations de Painlevé, 299-339, Sémin. Congr., 14, Soc. Math. France, Paris, 2006. 39. W. Wasow, Asymptotic expansions for ODE. Reprint of the 1965 edition. Robert E. Krieger Publishing Co., Huntington, N.Y., 1976.

  One uses (2.30) again and eventually concludes with lemma 2.3. 42. Y. Takei, An explicit description of the connection formula for the first Painlevé equation. In Toward the exact WKB analysis of differential equations, linear or non-linear (Kyoto, 1998), 204, 271-296, Kyoto Univ. Press, Kyoto, 2000.

).

  . The function H solves the fixed-point problem H = N(H)

		N(H) = M ρ,(0)	392 625	t 2 + 3t + 4t 2 H +	1 2	t 2 H 2 .	(3.25)
	We set M ρ,(0) = 1 ρ , U = D(0, ρ 4.22 ), and B ρ = {H ∈ O(U), H ≤ ρ}. One easily shows that for any H, H 1 , H 2 ∈ B 1 ,
	N(H) ∈ B ρ	and	N(H 1 ) -N(H 2 ) ≤	44150 44521	H 1 -H 2 .
	We conclude with the contraction mapping theorem: w(z) is holomorphic on the
	domain |z| >	4.22 ρ	and is bounded by ρ there. Therefore, by lemma 3.5, w is an
	entire function and satisfies: for every ξ ∈ C, | w(ξ)| ≤ 4.22e	4.22 ρ |ξ | . To sum up:
	Corollary 3.1. In theorem 3.3, one can choose A = 4.22 and τ = 4.22 ρ .
	with

  Berry, C.J. Howls, Hyperasymptotics for integrals with saddles. Proc. Roy. Soc. Lond. A 434 (1991), 657-675. 2. W. Balser, D.A. Lutz, R. Schäfke, On the convergence of Borel approximants. J. Dynam. Control Systems 8 (2002), no. 1, 65-92 3. E. Caliceti, M. Meyer-Hermann, P. Ribeca, A. Surzhykov, U.D. Jentschura, U. D. From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions. Phys. Rep. 446 (2007), no. 1-3, 1-96. 4. B. Candelpergher, C. Nosmas, F. Pham, Approche de la résurgence, Actualités mathématiques, Hermann, Paris (1993).

5. B. Candelpergher, C. Nosmas, F. Pham, Premiers pas en calcul étranger, Ann. Inst. Fourier (Grenoble) 43 (1993) 201-224. 6. O. Costin, On Borel summation and Stokes phenomena for rank-1 nonlinear systems of ordinary differential equations. Duke Math. J. 93 (1998), no. 2, 289-344. 7. O. Costin, Asymptotics and Borel summability, Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, 141. CRC Press, Boca Raton, FL, 2009.

  1. for each t ∈ I, the path H t : s ∈ I → H (s,t) is shortest symmetric; 2. the projection H 0 = p • H 0 of the initial path H 0 follows a segment in by Γ the endpoint path t ∈ I → H t (1), the product H 0 Γ , when reparametrized by arc-length parametrization, coincides with the shortest path from 0 to ζ .

	• R	(0) ρ ;
	3. denoting

Proof. Let ζ be a point in R

  . The reader is encouraged to check the remaining properties. Definition 4.19. Let ζ be any given point in R . The uniquely determined continuous map H given by proposition 4.3 is called the shortest symmetric (Z, ρ)homotopy associated with ζ . The path H 1 : s ∈ [0, 1] → H (s, 1) is called the shortest symmetrically contractile path associated with ζ in R Remark 4.3. Let H be a shortest symmetric (Z, ρ)-homotopy. Consider the path H t : s ∈ [0, 1] → H (s,t) for any given t ∈ [0, 1]. Then H t is the shortest symmetrically contractile path associated the endpoint H t (1) in R

		(1)
		(1) ρ and its length is denoted
	by leng ρ (ζ )
		(1) ρ .
	The next two statements are left as exercises.
	Lemma 4.8. The mapping ζ ∈ R	(1)

ρ

  This is one place where it is interesting to work with a "well-prepared" equation, see what we have done in Sect. 3.1 to get (3.6) and exercise 3.1):

	Lemma 5.6. If w 0

  We recall the following simple definition, for objects much discussed in[18]. Definition 7.38. The C-differential commutative and associative convolution algebra Cδ ⊕ RZ with unit δ , is called a space of Z-resurgent functions. We denote by RZ⊂ RES the C-differential commutative and associative convolution algebra made of resurgent singularities of the form ϕ = aδ + ϕ with ϕ ∈ R Z .Since Cδ ⊕ RZ is a convolution algebra, the identity ϕ * ϕ = ( ϕ * ϕ) (proposition 7.6) implies that RZ is indeed a convolution algebra. One usually uses abridged notation R in this course.Definition 7.39. A series expansion ϕ ∈ C[[z-1 ]] is a Z-resurgent series when its formal Borel transform B ϕ is a Z-resurgent function or, equivalently, when the asymptotic class ϕ belongs to RES Z . We denote by R Z the C-differential commutative and associative algebra made of Z-resurgent series.

	7.7.2.2 Resurgent functions, resurgent series
	) .(7.13)

  is the unique integral curve satisfying both dζ dt = X(ζ ,t) and the datum ζ (t 0 ) = ζ 0 . One finally sets φ t (ζ ) = g 0,t (ζ ). Notice that any integral curve ζ (t) of X has length less than length(γ) < R, since |X(ζ ,t)| ≤ |γ (t)|. With this remark and arguments detailed in

  Of course,(8.16) can be deduced also from(8.15) by formal Laplace transform (definition 7.25 and proposition 7.10).

	w 0 ) ∂ w	W .	(8.16)

1.2 On singularities of solutions of ordinary differential equations

Some elements about ordinary differential equations

This condition can be weakened by asking the general solution to be only uniformisable.

The first Painlevé equation

Just multiply both sides of the equality by v , then integrate.

The reader is referred to[16, 19] for details on asymptotic expansions.

Tritruncated solutions for the first Painlevé equation

A step beyond Borel-Laplace summability

Transseries and formal integral for the first Painlevé equation

Truncated solutions for the first Painlevé equation

Remember that a ke 1 ,0 = 0 as a rule, apart from the case k = 1 where a e 1 ,0 = 1.

One can also describe the solution in term of the Lambert function, the compositional inverse of the function xe x .

Supplements to resurgence theory

Modulo complex conjugation

As usual, modulo complex conjugation

We abide a notation of[1], although the construction therein slightly differs from ours.

That is H is a homotopy and for each t ∈ [0, 1], H t is an embedding. Remember that we see Γ I,ε as an embedded curve H 0 : R → C.

w e 2 ,i .

Acknowledgements I am indebted to my student Julie Belpaume for helping me to working out this chapter. I thank Jean Ecalle for interesting discussions on phenomena induced by resonances.

Acknowledgements I warmly thank my student Julie Belpaume who helped me to work out this chapter.

Appendix

The reader only interested in learning applications of resurgence theory may skip this appendix, where we show theorem 2.1 for completeness. We follow the proof given in [6]. See also [15,16] and specially [14] with comments and references therein. We start with two lemmas. Lemma 2.1. Let u be any solution of (2.1), holomorphic on a neighbourhood of x 0 ∈ C. Then the radius R of analyticity at x 0 satisfies R ≥ 1/ρ with is denoted by (R θ Z , p θ ). We often use abridged notation (R θ , p). We call principal sheet the unique domain R θ ,(0) ⊂ R θ so that the resctriction p| R θ ,(0) realizes a homeomorphism between R θ ,(0) and the simply connected domain • R θ ,(0) .

Resurgent functions

Various spaces of so-called resurgent functions can be defined and used according to the context. We start with the notion of resurgent singularities. 7.7.2.1 Resurgent singularities, resurgent asymptotic classes Definition 7. 35. A singularity ϕ ∈ SING is said to be Z-resurgent when it can be determined by a major ∨ ϕ ∈ ANA which can be analytically continued to C \ Z. We denote by RES Z or simply RES the space of Z-resurgent singularities.

A Z-resurgent singularity is often simply called a Z-resurgent function. Throughout this course we will usually write "resurgent singularity" in place of Z-resurgent singularity.

Remark 7.5. It is important to keep in mind that the minor ϕ of any resurgent singularity ϕ ∈ RES, can be analytically continued to C \ Z, since the minor ϕ does not depend on the chosen major. Definition 7.36. One says that ϕ ∈ RES is a resurgent constant when ϕ has a major which can be analytically continued to C. The space of resurgent constants is denoted by CONS. Definition 7.37. An asymptotic class ϕ ∈ ASYMP is called a Z-resurgent asymptotic class, resp. a resurgent constant, when its formal Borel transform ϕ is a Zresurgent singularity, resp. a resurgent constant. We denote by RES Z or simply RES the space made of Z-resurgent asymptotic classes. We denote by CONS the subspace of resurgent constants.

A Z-resurgent asymptotic class is often simply called a Z-resurgent function or even a resurgent function.

Example 7.8. The singularities I σ and Jσ,m are resurgent constants, as well as their associated asymptotic classes I σ and Jσ,m.

Before introducing the simply ramified resurgent functions, we need to state the following straightforward consequence of proposition 7.13. Lemma 7.9. The space SING s.ram of simply ramified singularities

Definition 7.56. One denotes by ASYMP s.ram the space of asymptotic classes associated with SING s.ram . The restriction of the Taylor map to ASYMP s.ram is denoted by T s.ram 1 . One denotes by s.ram its composition inverse, that is the natural extension of the mapping to

One can define the alien operators ∆ + ω , ∆ ω : R s.ram → R s.ram in the same manner than in definition 7.55 and, again, for any ϕ ∈ R s.ram , the alien derivative ∆ ω ϕ only depends on

• ω.

Ramified resurgent functions

The following definition makes sense by propositions 7.6 and 7.13.

through the Taylor map, for any given arc of S 1 .

We have encountered such formal expansions when we considered the formal integral for Painlevé I (theorem 5.1).

In the same way that C[[z -1 ]] 1 can be thought of as a constant sheaf on S 1 , the space (σ ,m)

] 1 can be seen as a constant sheaf on S 1 . This justifies the following definition. Definition 7.59. Let be θ ∈ S 1 and α > 0. We denote by Nils 1 , resp. Nils 1,(θ ,α) . the space of global sections of the sheaf (σ ,m)

The restriction of the Taylor map to ASYMP ram is denoted by T ram 1 . One denotes by ram :

its composition inverse, where J σ ,m (z) = (-1) m log m (z) z σ .

One can define the space Nils as well, made of formal expansions of the form

We can of course assume that for any i = j, σ iσ j / ∈ Z. We denote ω i = e -2iπσ i and we remark that ω iω j = 0 for any i = j. We set ρ. ϕ(z) = ϕ(ze 2iπ ) and more generally ρ k . ϕ(z) = ϕ(ze 2iπk )

. This implies that for each integer

ϕ. This observation can be generalized:

] are uniquely determined by ϕ and its monodromy (that is ρ. ϕ, ρ 2 . ϕ, etc.) once one imposes that

Proof. This is a well-known fact and we follow a reasoning from [20]. We only show how ϕ determines the series ϕ (σ i ,m) since we will use this result in a moment.

As a consequence, for any ϕ ∈ Nils one has P(ρ) ϕ ∈ Nils for any polynomial P ∈ C[X], and there exists a polynomial P ∈ C[X] such that P(ρ) ϕ = 0. We denote by d( ϕ) the degree of the minimal polynomial of the action of ρ on ϕ. We then make a reasoning by induction on d( ϕ). Suppose that d( ϕ) = 1. This means that there exists ω = e -2iπσ ∈ C such that

Suppose now that for any ϕ ∈ Nils such that d( ϕ) ≤ d, its decomposition is Suppose furthermore that ϕ is even, so that ρ -1/2 . ϕ(z) = ϕ(z). On deduces that

z σ e iπσ ψ(-z).

Comments

We mentioned in Sect. 4.7 the generalisation of the resurgence theory with the notion of "endlessly continuable functions" [1,8]. The whole constructions made in this chapter can be extended as well to this context. We of course owe the main ideas presented here from the work of Ecalle, who started his theory in the 1970's [6]. We have borrowed most of the materials to Pham et al. [1], in particular the microfunctions and the sheaf approach. To compare with other written papers devoted to resurgence theory, we have paid more attention to the sheaf and associated spaces of asymptotic classes. Finally, the responsability for possible mistakes is ours.

To what concerns the k-th series w

we have a result that ensues directly from theorem 6.1:

k belongs to R (1) , the asymptotic class W k = ram W k belongs to RES (1) and the singularity W k = B W k belongs to RES (1) .

where w [0] e i = δ + w e i for i =, 1, 2, otherwise

The main theorem

We formulate the main result of this chapter. We set ω j 1 = e 2iπ j (

with k 0 ∈ N , there exist two sequences of complex numbers A n (ω) n∈N and B n (ω) n∈N , uniquely determined by ω such that, for any k = (k 1 , k 2 ) ∈ N 2 and any n ∈ N,

resp.

(8.5)

where by convention

The sequences A n (ω) n∈N and B n (ω) n∈N are subject to the conditions: A n (ω) = 0 when |ω| ≥ n + 2 and B n (ω) = 0 when |ω| ≥ n + 1. Also, A n (ω) n∈N and B n (ω) n∈N are known for every ω ∈ C once they are known for arg ω = 0 only. In particular, A 0 (

The proof of this theorem will be given in Sect. 8.4.

Remarks

We detail (8.5) for n = 0. For any j ∈ Z and any k 0 ∈ N ,

and A 0 (k 0 ω j i ) = 0 when k 0 ≥ 2.

When k ∈ Ξ 1,0 , we use abridged notations

By proposition 7.28, ∆ ω

Remember that w 0 is even, thus w 0 = ρ -1/2 . w 0 , while w e 2 = ρ 1/2 . w e 1 . By proposition 7.28 again, ∆ ω j 2

ρ -1/2 . w 0 and we deduce that A 0 (ω j

2 ) = -iA 0 (ω j 1 ). Now for any k 1 ∈ N ,

and

and using (8.2),

By proposition 7.28, ∆ ω j 2

+ (-1) j B 1 (ω 0 2 )z -3/2 w e 1 and one deduces:

) . Of course, by symmetry: B 1 (ω j 1 ) = (-1) j B 1 (ω 0 1 ) + 2iπ jκ 2 A 0 (ω 0 1 ) . In the same way, ∆ ω j 2

+ iB 1 (ω j 1 )z -3/2 w e 1 and B 1 (ω j 2 ) = i B 1 (ω j 1 ) + iπκ 2 A 0 (ω j 1 ) . The resurgence coefficients are also called higher order Stokes cofficients in exponential asymptotics.

where, on the right-hand side, one recognizes the transseries solutions. The action of the symbolic Stokes automorphism allows to compare left and right Borel-Laplace summation: in their intersection domain of convergence,

This allows in particular to analytically continue the sum S θ j i -w 0 , thus the tritruncated solutions, onto a wider domain.

The same calculation can be made for the (convenient) transseries as well, and one easily gets, for i = 1, 2:

(8.11) Once again, this provides analytic continuations of the truncated solutions onto a wider domain.

It is a good place to mention medianization, since the ke i -th series w ke i are all real formal series. For instance, since w 0 belongs to R[[z -1 ]], its left and right Borel-Laplace sum are complex conjugate: S θ 0 1 + w 0 (z) = S θ 0 1 -w 0 (z). However, neither S θ 0 1 + w 0 nor S θ 0 1 -w 0 are real analytic functions, because of the Stokes phenomenon. The question is therefore the following one : can we construct from w 0 a real analytic function by a suitable morphism of differential algebras ?

The naive idea of taking their mean does not work (why ?).

The answer is "yes", by medianization or good averages, and is not unique. We refer to [24,14] for this question and its subtleties, see also [6].

Remark 8.1. The fact that the Stokes coefficient A 0 (ω 0 1 ) is nonzero can be deduced from the identity (8.10) : if A 0 (ω 0 1 ) = 0, then necessary the associated trituncated solution would be holomorphic on C \ K where K is a compact domain. This would mean that this trituncated solution has only a finite number of poles and that contradicts theorem 2.2. The fact that A 0 (ω 0 1 ) is pure imaginary can be seen also from (8.10) and from the realness of w 0 . For arg(z) = 0 and |z| large enough, one can write

and A 0 (ω 0 1 ) = -A 0 (ω 0 1 ) comes as an upshot.

As already said, the resurgent coefficients can be numerically calculated by hyperasymptotic methods. In return, resurgent coefficients and higher order Stokes phenomena play a crucial role in the hyperasymptotic approximations to Borel-Laplace sums, see for instance [9,26] and references therein.

The alien derivatives for the seen singularities

The idea that leads to theorem 8.1 relies on the following observations. We know by proposition 8.1 that the singularity W k belongs to RES (1) , for any k ∈ N 2 , and w 0 can be continued to an element of SING. This ends the proof of proposition 8.1 with the first approach.

Alien derivations, second approach

The second approach We now propose another approach, based on the notion of asymptotic classes, that uses tools akin to Gevrey and 1-summability theories.

We know that W = ∆ ω 0 1 w 0 ∈ ASYMP 0,π satisfies the condition P e 1 ( w 0 ) W = 0. We look at the equation P e 1 ( w 0 ) W = 0. The operator P e 1 ( w 0 ) is of order two in z and has two linearly independent formal solutions W e 1 = z Let us represent the asymptotic classes w 0 = w 0 , w e 1 = w e 1 and w e 2 = w e 2 on restriction to ASYMP 0,π . We pick a (good) open covering (I i ) of J =] -3π/2, 3π/2[ with open arcs I i of aperture less than π. We use the Borel-Ritt theorem for 1-Gevrey asymptotics to get, for each subscript i: w 0,i , w e 1 ,i , w e 2 ,i , ∈ A 1 (I i ) whose 1-Gevrey asymptotics is given by w 0 , w e 1 , w e 2 respectively. We know that each of these 1-Gevrey germ is uniquely defined up to 1-exponentially flat germs, that is up to elements of A ≤-1 (I i ). As a consequence, the collections (w 0,i ), (w e 1 ,i ), (w e 2 ,i ) represent the asymptotic classes we have in mind. For each subscript i, observe that

with D e 1 the linear operator given by definition 5.5, because the 1-Gevrey Taylor map T 1 (I i ) is a morphism of differential algebras. This ensures that D e 1 (w 0,i )w e 1 ,i belongs to A ≤-1 (I i ).

We draw a first conclusion : D e 1 ( w 0 ) w e 1 = 0 in ASYMP 0,π and thus, P e 1 ( w 0 ) W e 1 = 0 as well with W e 1 = z 3/2 w e 1 ∈ ASYMP 0,π .

We add a property that ensues from an analogue of the M.A.E.T. (theorem 3.1) and for which we refer to [14,16]: one can even find h i,e 1 ∈ A ≤-1 (I i ) so that D e 1 (w 0,i )(w e 1 ,ih e 1 ,i ) vanishes exactly, for each subscript i. Thus, one can find a representative w e 1 ,i ∈ A 1 (I i ) of w e 1 so that D e 1 (w 0,i )w e 1 ,i = 0 and thus, P e 1 (w 0,i )W e 1 ,i = 0 as well with W e 1 ,i = z 3 2 w e 1 ,i . The same reasoning yields: one can find a representative w e 2 ,i ∈ A 1 (I i ) of w e 2 so that D e 2 (w i,0 )w e 2 ,i = 0, thus P e 1 (w 0,i )e 2z w e 2 ,i = 0 with

k+n-e 2 e -λ λ λ .kz W k+n (8.34) Each component U k+n e -λ λ λ .(k+k 0 e 1 )z ∆ ω W k+n ∈ e -λ λ λ .(k+k 0 e 1 )z ASYMP arg(ω),π has its counterpart on the right-hand side of the equality. Necessarily,

This implies on the one hand hand that A n (ω) = 0 when |ω| ≥ n+2 while B n (ω) = 0 when |ω| ≥ n + 1. On the other hand,

.e 1 W k+m+k 0 e 1 (8.36)

).e 2 W k+m+k 0 e 1 with the convention used in theorem 8.1. The case

This result implies that the asymptotic class W k = ram W k belongs to RES (2) , as a consequence of corollary 7.3. An easy induction on k 0 ∈ N allows then to conclude that the W k belong to R Z ram . The rest of the theorem is shown by arguments used in remark 8.1.2.1. This ends the proof of theorem 8.1.

Comments

For differential systems of level 1 of the type (5.67), the resurgent study of the Stokes phenomenon and of the action of the symbolic Stokes automorphism ∆ / + θ on transseries solutions have been obtained by Costin [4], under some conditions. This has been later extended to more general differential equations (with no resonance), and also for difference equations of the type (5.68), in particular by Braaksma and his students (see [2,21]). These works make use of (so-called) "staircase distributions" [4,6] and do not make appeal to alien derivations. The method explained in this chapter is closer to the ideas of Ecalle, leading to the bridge equation and the full resurgent structure. Also, as we saw on the particular example of the first Painlevé equation, this method provides (theoretically) the whole set of Ecalle's holomophic invariants and passes the resonance cases under some conditions (no quasi-resonance, no nihilence [11]).

Index