Martin Klimeš

Confluence of singularities of non-linear differential equations via Borel-Laplace transformations

Keywords:

Confluence of singularities of non-linear differential equations via Borel-Laplace transformations

Introduction

When studying formal solutions of complex analytic ODE near a multiple singular point, it is the general rule to find divergent series. However, one can always construct true analytic solutions, defined on certain sectors attached to the singularity, which are asymptotic to the formal solution, and which are in some sense unique. In general, the solutions on different sectors do not coincide, and if extended to larger sectors, they may drastically change their asymptotic behavior due to the presence of hidden exponentially small terms, known as the (non-linear) Stokes phenomenon. In case where the singularity is a generic double point such sectoral solutions are obtained from the formal one using Borel-Laplace summation procedure. It is now understood, that the divergence of the formal asymptotic series is caused by singularities of its Borel transform, which also encode information on the geometry of the singularity. Another way how to understand the Stokes phenomena is by considering generic parameter depending deformations which split the multiple (irregular) singular point into several simple (regular) singularities: it turns out that the local analytic solutions at each singular point of the deformed equation in general do not match, thus explaining why solutions with nice asymptotic behavior at the limit when the singular points coalesce only exist in sectors.

When investigating families of analytic systems of ODEs depending on a complex parameter, that unfold a multiple singularity, one is faced with the problem that the Borel method of summation of formal series does not allow to deal with several singularities and their confluence. One of our goals here is to show how one can generalize (unfold) the Borel and Laplace operators in case of a generic singularity of multiplicity 2.

In this article we are investigating parametric families of first order non-linear differential systems unfolding a double singularity

(x 2 -) dy dx = M y + f (x, , y), (x, , y) ∈ C × C × C m , (1)
with M an invertible m × m-matrix, f (x, , y) an analytic germ, D y f (0, 0, 0) = 0, and where ∈ C is a small parameter. We study bounded parametric solutions of (1) near the singular points x = ± √ and their limits when → 0. Such solutions correspond to ramified center manifolds of an unfolded codimension 1 saddle-node singularity in a family of complex vector fields ẋ = x 2 -, ẏ = M y + f (x, , y).

For = 0, the divergence of the formal power series solution of [START_REF] Balser | Formal power series and linear systems of meromorphic ordinary differential equations[END_REF] means that an analytic center manifold does not exist. Instead there are "sectoral center manifolds" corresponding to the Borel sums of the divergent series.

For = 0, it is well known that for non-resonant values of the parameter there exists a local analytic solution on a neighborhood of each singularity x = ± √ . Previous studies of the confluence phenomenon [START_REF] Yu | On the Confluence phenomenon of Fuchsian equations[END_REF], [START_REF] Glutsyuk | Confluence of singular points and the nonlinear Stokes phenomena[END_REF] have focused at the limit behavior of these local solutions when → 0. Because the resonant values of accumulate at 0 in a finite number of directions, these directions of resonance in the parameter space could not be covered in those studies, except if the spectrum of M was of Poincaré type. Here we make no assumption other that M is invertible.

We will construct a new kind of parametric solutions of systems [START_REF] Balser | Formal power series and linear systems of meromorphic ordinary differential equations[END_REF] which are defined and bounded on certain ramified domains attached to both singularities x = ± √ (at which they possess a limit) in a spiraling manner. They depend analytically on √ from a sector of opening > π, thus covering a full neighborhood of the origin in the parameter space (including those parameters values for which the unfolded system is resonant), and they converge uniformly when → 0 to a pair of the classical sectoral solutions: Borel sums of the formal power series solution of the limit system, defined on two sectors covering a full neighborhood of the double singularity at the origin. In fact, each such pair of the sectoral Borel sums for = 0 unfolds to a unique above mentioned parametric solution.

We provide three different and complementary interpretations of these unfolded sectoral solutions: i) Using unfolded Borel and Laplace transformations: This is the principal approach of this article, with an advantage that it provides a unified treatment for all values of the parameter and explains the form of natural domains on which the solutions exist and are bounded. Most importantly, it gives an insight to intrinsic properties of the singularity and to the source of the divergence similar to that provided by the classical Borel-Laplace approach.

ii) Using the Hadamard-Perron theorem for = 0.

iii) Interpreting them as certain Borel sums of the unique formal power series in (x, ) solving [START_REF] Balser | Formal power series and linear systems of meromorphic ordinary differential equations[END_REF], which in turn is their asymptotic expansion. An important consequence of this correspondence is that the formal and the unfolded sectoral solutions satisfy the same {∂ x , ∂ }-partial differential relations.

These solutions were previously constructed by other methods in the special cases of dimension m = 1 and a general multiplicity of the singular point [START_REF] Rousseau | Analytical moduli for unfoldings of saddle-node vector filds[END_REF], and in the case of Riccati systems corresponding to normalizing transformations for families of linear differential systems unfolding a non-resonant irregular singularity [START_REF] Lambert | Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank 1[END_REF], [START_REF] Hurtubise | Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank k[END_REF], which motivated our present study. All our results translate directly to this situation, y playing role of such normalizing transformation (Section 2.3 below).

Statement of results

Notation

Borel-Laplace transformations and their unfolding

The Borel method of summation of (1-summable) divergent series is used to construct their sectoral Borel sums: unique analytic functions that are asymptotic to the series in certain sectors of opening > π at the singular point and satisfy the same differential relations.

Let ŷ(x) = +∞ k=1 y k x k be a formal power series. Using the Euler formula for the Γ-function: Γ(k) = +∞ 0 z k-1 e -z dz, equal to (k -1)! if k ∈ N >0 , one can write

x k = +∞e iα 0 ξ k-1 (k-1)! e -ξ
x dξ, for x in the half-plane e iα x > 0. Hence

ŷ(x) = +∞ k=1 y k x k = +∞ k=1 +∞e iα 0 y k (k -1)! ξ k-1 • e -ξ x dξ.
The formal Borel transform of ŷ is the series

B[ŷ](ξ) = +∞ k=1 y k (k -1)! ξ k-1 . (2)
If the coefficients of ŷ(x) have at most factorial growth ( |y k | ≤ c k k! for some c > 0), then the series B[ŷ](ξ) is convergent on a neighborhood of 0 with a sum φ(ξ). If moreover φ has an analytic extension to a half-line e iα R + and has at most exponential growth there (|φ(x)| ≤ K e Λ|ξ| , ξ ∈ e iα R + , for some K, Λ > 0), then its Laplace transform in the direction α

L α [φ](x) = +∞e iα 0 φ(ξ) • e -ξ x dξ (3)
is convergent for x in a small open disc of diameter 1 Λ centered at e iα 2Λ and extends to 0 (which lies on the boundary of the disc), defining there the Borel sum of ŷ(x) in direction α. A series ŷ[x] is Borel summable (or 1-summable) if its Borel sums L α [ B[ŷ]](x) exist in all but finitely many directions 0 ≤ α < 2π. When varying continuously the direction in which the series is summable, the corresponding Borel sums are analytic extensions one of the other, yielding a function defined on a sector of opening > π.

Let us remark that ŷ[x] is convergent if and only if it is Borel summable in all directions. This means that the Borel sums of divergent series can only exist on sectors. This is also known as the Stokes phenomenon.

The Borel sums of ŷ(x) are asymptotic of Gevrey order 1 to the formal series ŷ(x) at the origin, and most importantly, they satisfy the same analytic differential equations as ŷ(x). More detailed information on the Borel summability can be found, for example, in [START_REF] Martinet | Théorie de Galois differentielle et resommation[END_REF], [START_REF] Malgrange | Sommation des séries divergentes[END_REF] or [START_REF] Balser | Formal power series and linear systems of meromorphic ordinary differential equations[END_REF].

A typical source of Borel summable power series are formal solutions of generic ODEs at an irregular singular point of multiplicity 2.

Example 1. A non-homogeneous linear analytic ODE with a double singularity at the origin

x 2 dy dx = y + f (x), (x, y) ∈ C × C, (4)
where f (x) ∈ x C{x} is a convergent power series, possesses a unique formal solution ŷ(x). Generically, this series is divergent (for instance if f (x) = -x then ŷ(x) = +∞ n=1 (n -1)!x n is the Euler series). The formal Borel transform of the equation ( 4)

is ξ • B[ŷ](ξ) = B[ŷ](ξ) + B[f ](ξ),
hence the reason for the divergence of ŷ(x) is materialized by the singularity of

B[ŷ](ξ) = B[f ](ξ) ξ-1 at ξ = 1. The Borel sum y(x) = L α [ B[ŷ]](x) of ŷ(x), α ∈ ]0, 2π
[, is a solution to (4), well defined in a ramified sector arg x ∈ ] -π 2 + η, 5π 2 -η[ for any η > 0. The set (x, y(x)) is a center manifold of a saddle-node singularity of the vector field ẋ = x 2 , ẏ = y + f (x).

This example shows that in general an analytic center manifold does not exist, but instead there are "sectoral center manifolds".

The inverse to the Laplace transformation in direction α is the analytic Borel transformation: If y(x) is a germ of function analytic on a closed sector of opening ≥ π bisected by e iα R + that vanishes at 0 as O(x λ ) uniformly in the sector for some λ > 0, then its analytic Borel transform in direction α is defined as the "Cauchy principal value" (V.P.) of the integral

B α [y](ξ) = 1 2πi V.P. γ y(x) e ξ x dx x 2 , for ξ ∈ e iα R, (5)
over a circle γ = { e iα x = C}, C > 0, inside the sector. The plane of ξ is also called the Borel plane.

The formal Borel transform (2) of an analytic germ y vanishing at 0 is related to the analytic one by

B α [y](ξ) = χ + α (ξ) • B[y](ξ), for ξ ∈ e iα R, (6)
where

χ + α (ξ) = 1, if ξ ∈ ]0, +∞e iα [, 0, if ξ ∈ ]-∞e iα , 0[.
The idea of unfolding the Borel-Laplace operators in order to generalize the methods of Borel summability and resurgent analysis to systems with several confluent singularities was initially brought up by Sternin and Shatalov in [START_REF] Yu | On the Confluence phenomenon of Fuchsian equations[END_REF]. The key lies in appropriate unfolding of the "kernels" e ξ x dx x 2 and e -ξ

x dξ of the transformations (3) and ( 5), and in right determination of the paths of integration. The Borel transformation is designed so that it converts the derivation x 2 d dx to multiplication by ξ, and we will want to preserve this property.

The complex vector field x 2 ∂ ∂x with a double singularity at the origin is naturally (and universally) unfolded as

(x 2 -) ∂ ∂x , ∈ C. (7)
We will associate to it the unfolded Borel and Laplace transformations

B + α [y](ξ, √ ) = 1 2πi V.P. e iα t(x, )=C y(x) e t(x, )ξ dt(x), 0 < C < e iα πi √ , L α [φ](x, √ ) = +∞e iα -∞e iα φ(ξ) e -t(x, )ξ dξ, (8)
where

t(x, ) = - dx x 2 - := -1 2 √ log x- √ x+ √ , if = 0, 1 x , if = 0, (9)
is the negative complex time of the vector field [START_REF] Costin | On Borel summation and Stokes phenomena for rank-1 nonlinear systems of ordinary differential equations[END_REF]. Let us remark that the (unilateral) Laplace transformation L α [φ] (3) is equal to the (bilateral) Laplace transformation L α [φ] with = 0 and t(x, 0) = 1 x , if one extends the integrand by 0 for ξ ∈ ]-∞e iα , 0[ :

L α [φ](x) = L α [χ + α φ](x, 0
). In Sections 3 and 4 we will establish some general properties of these transformations based on the classical theory of Fourier and Laplace integrals, and in Section 5 we will apply them to study solutions of (1) in the vicinity of the singular points.

Center manifold of an unfolded codimension 1 saddle-node type singularity

We consider an analytic family of vector fields in C m+1 unfolding such saddle node singularity in form

ẋ = x 2 -, ẏ = M y + f (x, , y), (x, , y) ∈ C × C × C m , (10)
with ∈ C a small parameter, M invertible, and f a germ of analytic vector function at the origin of

C × C × C m with D y f (0, 0, 0) = 0, and f (x, , 0) = O(x 2 -). 1 (11)
For = 0 the vector field [START_REF] Glutsyuk | Confluence of singular points and the nonlinear Stokes phenomena[END_REF] possesses a ramified 1-dimensional "center manifold" consisting of several sectoral pieces tangent to the x-axis. Here we study its parametric unfolding in the family: It is given as a graph of a function y = y(x, √ ), ramified at x = ± √ , satisfying the singular non-linear system of m ordinary differential equations

(x 2 -) dy dx = M y + f (x, , y), (x, , y) ∈ C × C × C m . ( 12
)
Remark 2. If f (x, , 0) = 0, then [START_REF] Hurtubise | Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank k[END_REF] has a unique analytic solution given by y = 0. Being trivial, this case bears no interest in this article. Reciprocally, if [START_REF] Hurtubise | Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank k[END_REF] has an analytic solution y = φ(x, ), then the change of variable y → y -φ(x, ) brings [START_REF] Glutsyuk | Confluence of singular points and the nonlinear Stokes phenomena[END_REF] to a form with f (x, , 0) = 0.

Remark 3. In dimension m = 1, families of vector fields unfolding a saddle-node of codimension k were thoroughly studied in [START_REF] Rousseau | Analytical moduli for unfoldings of saddle-node vector filds[END_REF].

Remark 4. A general analytic family of vector fields in C m+1 unfolding a saddlenode singularity of codimension 1 to two simple singularities is locally orbitally analytically equivalent to

ẋ = (x 2 -) + G(x, , y), ẏ = M y + F (x, , y), (13)
with G(x, , y) = o(|y|), F (x, , y) = O(x 2 -) + o(|y|). The singular transformation (blow-up) y = (x 2 -)u brings [START_REF] Ilyashenko | Lectures on Analytic Differential Equations[END_REF] to

ẋ = (x 2 -) 1 + G(x, ,(x 2 -)u) (x 2 -) , u = M u + F (x, ,(x 2 -)u) (x 2 -) -2x 1 + G(x, ,(x 2 -)u) (x 2 -) u,
which by the assumption is analytic near 0 ∈ C × C × C m . A transformation sending its two singularities to the points (x, u) = (± √ , 0) and a division by a non-vanishing germ reduces it to the from (10). 2 M u ± √ + f (± √ , u ± √ , ), with u±0 = 0, then the change of variable y → y -

1 2 √ u + √ (x+ √ ) - u -√ (x- √
) , analytic in (x, ), would bring the system (10) to a one with f (x, , 0) = O(x 2 -). 2 For m = 1, it's been shown in [START_REF] Rousseau | Analytical moduli for unfoldings of saddle-node vector filds[END_REF]Proposition 3.1], cf. also [10, Lemma 1], that the the family [START_REF] Ilyashenko | Lectures on Analytic Differential Equations[END_REF] is in fact locally orbitally analytically equivalent to a family [START_REF] Glutsyuk | Confluence of singular points and the nonlinear Stokes phenomena[END_REF].

Formal solution.

Proposition 5 (Formal solution). The system (12) possesses a unique solution in terms of a formal power series in (x, ):

ŷ(x, ) = +∞ k,j=0 y kj x k j , y kj ∈ C m . ( 14
)
This series is divisible by (x 2 -), and its coefficients satisfy y kj ≤ L k+2j (k+2j-1)! for some L > 0.

Proof. Write ŷ(x, ) = (x 2 -) k,j u kj x k j , u kj ∈ C m , f (x, , y) = |l|≥0 k,j f l,k,j x k j y l , f l,kj ∈ C m ,
where y l := y l 1 1 • . . . • y lm m for each multi-index l = (l 1 , . . . , l m ) ∈ N m , and |l| = l 1 + . . . + l m . Substituting ŷ(x, ) for y in f , dividing the equation ( 12) by (x 2 -), and comparing the coefficients of x k j , one obtains a set of equations

M u kj = (k+1)(u k-1,j -u k+1,j-1 ) + P kj ,
where P kj is a polynomial in {u k j | k ≤ k, j ≤ j, k +2j < k+2j}. By recursion with respect to the order k + 2j of the indices (k, j) this uniquely determines all the coefficient vectors u kj , and therefore also the coefficient vectors y kj = u k-2,j -u k,j-1 .

Similarly, y kj (note that k + 2j ≥ 2) satisfy recursive equations

M y kj = (k+1)y k+1,j-1 -(k-1)y k-1,j + Q kj ,
where Q kj is a polynomial in {y k j | k ≤ k, j ≤ j, k + 2j < k + 2j}, given by

Q kj = -f 0,k,j - |l|=1 κ≤k ι≤j 2≤κ+2ι<k+2j f l,k-κ,j-ι a l,κ,ι - |l|≥2 κ≤k ι≤j 2|l|≤κ+2ι f l,k-κ,j-ι a l,κ,ι ,
where a l,κ,ι are polynomials in {y k j | k ≤ κ, j ≤ ι, k + 2j ≤ κ + 2ι -2|l|} determined by (ŷ(x, )) l = κ+2ι≥2|l| a l,κ,ι x κ ι . We want to show by induction on the order k + 2j that y kj ≤ L k+2j (k+2j-1)!, for some L > 0 independent of (k, j), for • the maximum norm on C m .

Let us first show that if

y k j ≤ L k +2j (k + 2j -1)! for each k ≤ κ, j ≤ ι, k +2j ≤ κ+2ι-2|l|, then a l,κ,ι ≤ L κ+2ι 2 |l|-1 (κ+2ι-|l|)! |l|! , 2|l| ≤ κ + 2ι. This is certainly true if |l| = 1. If |l| ≥ 2, then a l,κ,ι ≤ k ,j a l ,κ-k ,ι-j • y k ,j ,
where l is a smaller multi-index, |l | = |l| -1, and the sum is taken over the set of indices {(k , j

) | k ≤ κ, j ≤ ι, 2 ≤ k +2j ≤ κ+2ι-2|l|+2} whose cardinality is equal (κ + 1)(ι + 1) -|l|(|l| + 1) -2 < (κ + 2ι -|l| -1)(κ + 2ι -|l|). We can write (κ+2ι-k -2j -|l|+1)! • (k +2j -1)! ≤ 2 |l| (κ+2ι-|l|-2)! since the first factorial is at least (|l|-1)!. Hence the estimate. Now if f l,k,j ≤ L k+2j 1 L |l| 2 for some L 1 , L 2 > 0, then Q kj L -k-2j (k+2j-1)! ≤ L 1 L k+2j + mL 2 k+2j-1 κ,ι (κ+2ι-1)! (k+2j-2)! L 1 L k+2j-κ-2ι + + k+2j 2 p=2 (2mL 2 ) p p!(k+2j-p+1)...(k+2j-1) κ,ι (κ+2ι-p)! (k+2j-p)! L 1 L k+2j-κ-2ι .
For L > L 1 , each of the two sums over (κ, ι) can be estimated from above by 3, hence, taking into account that k

+ 2j -p ≥ k+2j 2 , Q kj L -k-2j (k+2j-1)! ≤ L 1 L k+2j + 3 mL 2 k+2j-1 + 3 (2mL 2 ) 2 k+2j-1 k+2j 2 p=2 1 p! 2mL 2 k+2j 2 +1 p-2
, which can be made less than

(k + 1)y k+1,j-1 -(k -1)y k-1,j ≤ 2L -k-2j-1 • (k + 2j - 1)!.

Sectoral center manifold and its unfolding.

For = 0 it is known that the equation ( 12) has a unique solution in terms of a Borel summable formal power series ŷ0 (x) = ŷ(x, 0) [START_REF] Lambert | Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank 1[END_REF]. This a very classical theorem going essentially back to J. Horn, M. Hukuhara or J. Malmquist [START_REF] Malmquist | Sur l'étude analytique des solutions d'un système d'équations différentielles dans le voisinage d'un point singulier d'indétermination[END_REF] among many, whose modern versions can be found for example in works of Braaksma [START_REF] Braaksma | Laplace integrals in singular differential and difference equations[END_REF], Écalle, or Ramis and Sibuya [START_REF] Ramis | Hukuhara domains and fundamental existence and uniqueness theorems for asymptotic solutions of Gevrey type[END_REF], (see also [START_REF] Martinet | Problèmes de modules pour des équations différentielles non linéaires du premier ordre[END_REF] for the case m = 1): Theorem 6. For = 0 the formal solution ŷ0 (x) = +∞ k=0 y k0 x k of the system (10) is Borel summable in each direction α with e iα R + disjoint to Spec M . Hence to each connected component Ω of C λ∈Spec M λR + in the Borel plane (Figure 1) corresponds a unique Borel sum of ŷ0 (x), a solution of the system, defined on a sector in the x-plane of opening > π and asymptotic to ŷ0 (x).

More generally, for each j ∈ N, the formal component ŷj (x) := +∞ k=0 y kj x k of (14) is Borel summable in the same directions.

Proof. The Borel summability of ŷj (x) is obtained by recursion on j using Theorem 4 in [START_REF] Braaksma | Laplace integrals in singular differential and difference equations[END_REF]: each ŷj (x) is a formal solution to a system of differential equations x 2 dy j dx = M y j + h j (x) + g j (x, y j ), g j (x, y j ) = O(x y j ), with h j , g j depending polynomially on ŷ0 (x), . . . , ŷj-1 (x), thus Borel summable in x in directions disjoint to Spec M .

This means that for each two opposite components Ω + , Ω -in the Borel plane (i.e. such that Ω + ∪ Ω -∪ {0} contains some straight line e iα R), the two corresponding sectors X ± 0 of summability form a covering of a neighborhood of the origin in the x-plane. Theorem 9 below shows that each such covering pair of sectors {X + 0 , X - 0 } unfolds for = 0 to a single ramified domain X( √ ), adherent to both singular points x = ± √ (see Figure 3), on which there exists a unique bounded solution y(x, √ ) of ( 12), depending analytically on √ taken from a sector S of opening > π, that converge uniformly to the two respective Borel sums of ŷ0 (x) on X + 0 , X - 0 , when √ → 0. First we construct these domains.

Definition 7 (Family of domains X( √ ), √ ∈ S). Let {Ω + , Ω -} be a pair of opposite sectoral components of C λ∈Spec M λR + , and let

β 1 < β 2 be such that α∈ ]β 1 ,β 2 [ e iα R ⊆ Ω + ∪ Ω -∪ {0}. For some 0 < η < β 2 -β 1
4 , ρ > 0 and Λ > 0, let

S = { √ ∈ C | arg √ ∈ ]β 1 -π + 2η, β 2 -2η[, | √ | < ρ} ∪ {0}, (15)
and for each √ ∈ S let

T + α (Λ, √ ) :={Λ < (e iα t) < -( e iα πi √ ) -Λ}, T - α (Λ, √ ) :={-Λ > (e iα t) > ( e iα πi √ ) + Λ}, (16)
be slanted strips in the time t-plane in direction -α + π 2 that pass in between closed discs of radius Λ centered at the points 0 and ∓ πi √ . Define

T ± ( √ ) = α T ± α (Λ, √ )
(see Figure 2) as their union with varying

α 3 max{arg √ , β 1 } + η < α < min{β 2 , arg √ + π} -η. ( 17
)
Figure 2: The domains T ± ( √ ) in the time t-plane (9) according to √ ∈ S. (Here We define the domains X ± ( √ ) (see Figure 3) as simply connected ramified projections of T ± ( √ ) to the x-coordinate4 by the map

β 1 ∼ π 4 , β 2 ∼ 3π 4 , η ∼ 0).
t → x(t, ) = √ 1+e -2 √ t 1-e -2 √ t = √ coth √ t, if = 0, 1 t , if = 0, (18)
the inverse of ( 9), to which we adjoin the ramification points { √ , -√ } (which are approached from within the interior of X( √ ) following logarithmic spirals). For

√ = 0, X + ( √ ) = X -( √ ) =: X( √ ). On the other hand, X + (0) is a sectoral domain with arg x ∈ ]β 1 -π 2 + η, β 2 + π 2 -η[, while X -(0)
is its opposite -we define X(0) as their ramified union with 0 as the only common point.

Clearly, the domains X( √ ) depend continuously on √ ∈ S {0} and they converge, when √ → 0 radially with arg √ = β, to a pair of sub-domains X ± β (0) of X ± (0), and X ± (0) is the union of all these radial limits.

If the radius ρ of S is taken small enough, then there exists a fixed neighborhood of the origin in the x-plane covered by each domain

X( √ ), √ ∈ S.
Remark 8. The ramified domains X( √ ) are swept by complete real trajectories of the complex vector fields e i( πi 2 -α) (x 2 -) ∂ ∂x , with α as in [START_REF] Martinet | Problèmes de modules pour des équations différentielles non linéaires du premier ordre[END_REF], that stay forever within a neighborhood of 0 of radius ∼ 1 Λ , and tend to the point

√ (resp. - √ ) in negative (resp. positive) time.
Theorem 9. Consider a system [START_REF] Hurtubise | Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank k[END_REF] with M an invertible m×m-matrix and f (x, , y) as in [START_REF] Glutsyuk | Normal forms, bifurcations and finiteness problems in differential equations[END_REF]. (i) Let {Ω + , Ω -} be a pair of opposite components of C λ∈Spec M λR + (i.e. such that Ω + ∪ Ω -∪ {0} contains some straight line e iα R). For any arbitrarily small angle ν > 0 there are Λ, ρ > 0, such that on the corresponding family of domains X( √ ), √ ∈ S, of Definition 7, there is a unique bounded analytic solution y(x, √ ) to [START_REF] Hurtubise | Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank k[END_REF]. It is uniformly continuous on

X = {(x, √ ) | x ∈ X( √ )}
and analytic on the interior of X, and it vanishes (is uniformly O(x 2 -)) at the singular points. When √ tends radially to 0 with arg √ = β, then y(x, √ ) converges to y(x, 0) uniformly on compact sets of the sub-domains X ± β (0) ⊆ X ± (0), and the restriction of y(x, 0) to X ± (0) is the Borel sum of the formal series ŷ(x, 0) [START_REF] Lambert | Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank 1[END_REF] in directions of Ω ± .

The solution y(x, √ ), and its domain X, associated to each pair {Ω + , Ω -} are unique up to the reflection (x, √ ) → (x, -√ ), or analytic extensions. (ii) If, moreover, the spectrum of the matrix M is of Poincaré type (the convex hull of Spec M does not contain 0 inside or on the boundary), i.e. if there exists a (unique) component

Ω 1 of C λ∈Spec M λR + of opening > π, then the solution y 1 (x, √ ) on the domain X 1 ( √ ), √ ∈ S 1 , associated to the pair {Ω 1 , Ω 1 }
is ramified only at one of the singular points, and analytic at the other (see Figure 4).

Such is the case in dimension m = 1. The solutions y(x, √ ) will be constructed in Section 5 in form of two-sided Laplace integrals

y(x(t, ), √ ) = +∞e iα -∞e iα υ ± (ξ, √ ) e -tξ dξ, t ∈ T ± α (Λ, √ ),
with υ ± a solution to a non-linear convolution equation (corresponding to [START_REF] Hurtubise | Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank k[END_REF] via the unfolded Borel/Laplace transformations ( 8)) on strips in the Borel plane, which will be obtained using a fixed-point argument.

Proposition 10. Let y(x, √ ) be the solution of Theorem I, let √ ¯ , √ ˜ = e πi √ ¯ ∈ S be two opposite roots of as in Figure 2, and let ȳ± (t, √

) (resp. ỹ± (t, √ )) be the lifting of y(x, √ ¯ ) on T ± ( √ ¯ ) (resp. y(x, √ ˜ ) on T ± ( √ ˜ )). There are constants 0 < R < min λ∈Spec M |λ| and C 1 , C 2 > 0 independent of , such that for every t ∈ T ± ( √ ˜ ) ∩ T ∓ ( √ ¯ ), see Figure 5, ỹ± (t, √ ) -ȳ∓ (t, √ ) ≤ C 1 + C 2 d(t, ) e -Rd(t, ) , (19)
where d(t, ) denotes the distance of t from the border of The linearization of the vector field [START_REF] Glutsyuk | Confluence of singular points and the nonlinear Stokes phenomena[END_REF]

T ± ( √ ˜ ) ∩ T ∓ ( √ ¯ ).
at x = ± √ is equal to ẋ = ±2 √ (x ∓ √ ), ẏ = M ± √ y, M ± √ = M + O( √ ). (20)
(i) Let a line e iα R separate the point 2 √ and k of the eigenvalues of M from the point -2 √ and the other m -k eigenvalues (0 ≤ k ≤ m), see Figure 6. Then for small enough, the respective eigenvalues of M ± √ lie on the same sides of the line e iα R, hence by the Hadamard-Perron theorem [START_REF] Ilyashenko | Lectures on Analytic Differential Equations[END_REF] the vector field (10) has a unique (k + 1)-dimensional local invariant manifold at ( √ , 0) tangent to the x-axis and the corresponding k eigenvectors of M √ , and a unique (m -k + 1)-dimensional local invariant manifold at (-√ , 0) tangent to the x-axis and the corresponding m -k eigenvectors of M - √ . They intersect transversally as the graph of the solution y(x, √ ) of Theorem 9. Since the root parameter √ can vary as long as ±2 √ stay in their respective half-planes bounded by e iα R, whose angle α can also vary a bit, this gives a sector S of opening > π. We see that one cannot continue this description in √ beyond such maximal sector S.

(ii) If all the eigenvalues of M are in a same open sector of opening < π (i.e. Spec M is of Poincaré type), and -2 √ lies in the interior of the complementary sector Ω 1 of opening > π, then one obtains the solution y 1 (x, √ ) from Theorem 9 as a continuation of the local analytic solution at x = -√ (i.e. of the local invariant manifold of (10) tangent to the x-axis, provided by the Hadamard-Perron theorem) to the domain X 1 ( √ ). While this Hadamard-Perron approach explains where do the solutions of Theorem 9 come from, it does not provide their natural domain on which they are bounded. One should however notice the similarities between the description provided by the Hadamard-Perron theorem for = 0 (Figure 6) and that of the Borel summation for = 0 (Figure 1). In Section 5 we will unify the two of them using the unfolded Borel-Laplace transformations.

Remark 11 (Local invariant manifolds for non-resonant = 0 and their convergence). If the simple singular point of [START_REF] Glutsyuk | Confluence of singular points and the nonlinear Stokes phenomena[END_REF]

at x = √ = 0 satisfies the following non-resonance condition 2 √ N ∩ Spec M √ = ∅,
then it is known that the equation ( 12) possesses a unique convergent formal solution near x = √ , i.e. the vector field (10) has a 1-dimensional local analytic invariant manifold tangent to the x-axis at the singularity. The resonant values √ = λ 2n , λ ∈ Spec M √ , n ∈ N >0 , accumulate at the origin along the rays λR + , λ ∈ Spec M , dividing the √ -plane in a finite number of sectors (Figure 7). The following theorem was proven by A. Glutsyuk [START_REF] Glutsyuk | Confluence of singular points and the nonlinear Stokes phenomena[END_REF].

Theorem 12 (Glutsyuk). If √ = 0 lies inside one of these sectors (i.e. √ R + ∩ Spec M = ∅), then the local analytic solution at x = √ converges, when √ tends radially to 0, to the sectoral Borel sum L α [ B[ŷ 0 ]](x) of the formal solution of the limit system (cf. Figure 1), where α = arg √ (this is the direction on which lies the corresponding eigenvalue 2 √ of the linearization (20)).

Unless the spectrum of M is of Poincaré type, these sectors in the √ -plane on which the convergence happens are of opening < π.

Asymptotic expansions

Inner asymptotic expansion. Blowing-up the x-coordinate let z = x √ and

Y (z, √ ) := y( √ z, √ ), √ z ∈ X( √ ),
be the solution of Theorem 9, and

ŷ( √ z, √ ) =: Ŷ (z, √ ) = +∞ p=2 Y p (z) √ p , Y p (z) = 0≤k≤p k+2j=p y kj z k (21)
the formal solution [START_REF] Lambert | Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank 1[END_REF]. Since, √ t(x, ) = t(z, 1), ) by its uniqueness, and therefore it is also its Borel sum on S (the opening of S is > π). We need yet to specify the domain of z on which this is true. First, remark that

t(x, ˜ ) ∈ T ± ( √ ¯ ) ∩ T ∓ ( √ ˜ ) = T ± β 2 -η (Λ, √ ˜ ) ∩ T ± β 1 +η (Λ, √ ˜ ),
where β 1 , β 2 , η are as in Definition 7, and √ ˜ ∈ S ∩ e πi S, √ ¯ = e -πi √ ˜ . Hence t(z, 1) belongs to a limit of such rhomboidal domains scaled by √ ˜ , as √ ˜ → 0 radially with arg √ ˜ = β + π 2 (β being the direction of the Borel summation):

t(z, 1) ∈ T ± β 2 -η-β-π 2 (0, 1) ∩ T ± β 1 +η-β-π 2 (0, 1). ( 22
)
In the z-coordinate, z = coth t(z, 1), this corresponds to the limit of central region of the intersection

1 √ ˜ X( √ ˜ ) ∩ 1 √ ¯ X( √ ¯ )
, which covers a neighborhood of the origin and extends towards ∞ as a double sector arg z ∈ ∓ π Returning back to the x-coordinate and using slightly modified version of the Borel-Laplace summation operators, following [START_REF] Balser | Summability of power series in several variables, with applications to singular perturbation problems and partial differential equations[END_REF], we obtain:

2 -β + ]β 1 + η, β 2 -η[. Proposition 13 (Inner asymptotic expansion). The blow-up Y (z, √ ) = y( √ z, √ ) of the sectoral solution of Theorem 9 is equal to the Borel sum in √ of the blown-up formal solution Ŷ (z, √ ) = +∞ p=2 Y p (z) √ p , for √ ∈ S {0} and z ∈ 1 √ X( √ ) satisfying (
Theorem 15 (Borel sum of ŷ(x, √
)). Let U (x, ) be analytic extension of the function given by the convergent series in (x, )

U (x, ) = j,k y kj (k + 2j)! x k j .
For each point (x, √ ), for which there is an angle

θ ∈ ] -π 2 , π 2 [ such that the set S θ • (x, √
) ⊆ X, with S θ ⊂ C denoting the circle through the points 0 and 1 with center on e iθ R + , we can express y(x, √ ) as the following Laplace transform of U :

y(x, √ ) = +∞e iθ 0 y(x, ) = Y (z, √ ) = 1 √ +∞e iα 0 U (νz, ν 2 ) e -ν √ dν = +∞e iθ 0 U (sx, s 2 ) e -s ds after substitution ν = √ s. For √ = 0, U (ξ, 0) = B[x ŷ(x, 0)](ξ) w.r.t.
x, thus one obtains [START_REF] Schäfke | Confluence of several regular singular points into an irregular singular one[END_REF] by substituting ξ = sx in the classical Laplace transform (3) of U (ξ, 0). Moreover and the condition on the points (x, 0) ∈ X is satisfied for all x ∈ X ± (0).

Remark 16. The Borel-Laplace summation of Theorem 15 preserves algebraic operations as well as differentiation with respect to x, [START_REF] Balser | Summability of power series in several variables, with applications to singular perturbation problems and partial differential equations[END_REF]. This means that the sectoral solution y(x, √ ) and the formal solution ŷ(x, ) satisfy the same polynomial { ∂ ∂x , ∂ ∂ }-differential relations over the ring C{x, } of convergent series.

Outer asymptotic expansion. Let ŷ(x, ) = +∞ k,j=0 y kj x k j be the formal solution [START_REF] Lambert | Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank 1[END_REF] and let y ± j (x), j ∈ N, be Borel sums of ŷj (x) = +∞ k=0 y kj x k provided by Theorem 6 on the domains X ± (0). One can then consider the formal series in

+∞ j=0 y ± j (x) j . ( 24
)
It has been shown in [START_REF] Parise | Confluence de singularités régulieres d'équations différentielles en une singularité irréguliere[END_REF] (in the case of normalizing transformations for non-resonant irregular linear systems, cf. Section 2.3 below) that the sectoral solution y(x, √ ) of Theorem 9 is asymptotic to [START_REF] Sibuya | Gevrey property of formal solutions in a parameter[END_REF] of Gevrey order 1 in √ on a sector S ⊂ S on which both singularities { √ , -√ } are inside the same domain X ± (0), i.e. √ ∈ X + (0) ∩ X -(0). We call [START_REF] Sibuya | Gevrey property of formal solutions in a parameter[END_REF] an outer asymptotic expansion as it is defined for x in an "outer" region, |x| > | √ |.

Remark 17. The inner [START_REF] Rousseau | Modulus of orbital analytic classification for a family unfolding a saddlenode[END_REF] and outer [START_REF] Sibuya | Gevrey property of formal solutions in a parameter[END_REF] asymptotic expansions corresponding to ŷ(x, ) can be seen as a special case of a composite asymptotic expansion with a trivial fast part in the sense of [START_REF] Fruchard | Composite Asymptotic Expansions[END_REF], with the small difference that the functions y ± j (x) ∼ +∞ k=0 y kj x k are sectoral, rather than analytic on a fixed neighborhood of 0.

Sectoral normalization of families of non-resonant linear differential systems

An application of Theorem 9, interesting on its own, is the problem of existence of normalizing transformations for linear differential systems near an unfolded nonresonant irregular singularity of Poincaré rank 1. We will show that this problem can be reduced to a system (12) of m = n (n -1) Ricatti equations (where n is the dimension of the system), providing thus a proof of a sectoral normalization theorem by Parise [START_REF] Parise | Confluence de singularités régulieres d'équations différentielles en une singularité irréguliere[END_REF], Lambert and Rousseau [START_REF] Lambert | Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank 1[END_REF]. Consider a parametric family of linear systems ∆(x, ) y = 0 given by

∆(x, ) = (x 2 -) d dx -A(x, ), (x, ) ∈ (C × C, 0) (25)
where y(x, ) ∈ C n , A(x, ) is analytic, and assume that the eigenvalues λ

(0) i (0), i = 1, . . . , n, of the matrix A(0, 0) are distinct. Let λ i (x, ) = λ (0) i ( ) + xλ (1)
i ( ), i = 1, . . . , n, be the eigenvalues of A(x, ) modulo O(x2 -), and define

∆(x, ) = (x 2 -) d dx -Λ(x, ), Λ(x, ) = Diag(λ 1 (x, ), . . . , λ n (x, )), (26)
the formal normal form for ∆. The problem we address, is to find a bounded invertible linear transformation y = T (x, √ ) u between the two systems ∆y = 0 and ∆u = 0. Such T is a solution of the equation

(x 2 -) dT dx = AT -T Λ. ( 27
)
Note that if V (x, ) is an analytic matrix of eigenvectors of A(x, ) then the transformation y = V (x, ) y 1 brings the system ∆y = 0 to ∆ 1 y 1 = 0, whose matrix is written as

A 1 (x, ) = Λ(x, ) + (x 2 -)R(x, ), with R = -V -1 dV
dx . Hence we can suppose that system [START_REF] Yu | On the Confluence phenomenon of Fuchsian equations[END_REF] is already in such form. The following theorem is originally by Parise [START_REF] Parise | Confluence de singularités régulieres d'équations différentielles en une singularité irréguliere[END_REF], and by Lambert and Rousseau [START_REF] Lambert | Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank 1[END_REF]Theorem 4.21], generalizing earlier investigations by Zhang [START_REF] Zhang | Confluence et phénomène de Stokes[END_REF] 5 of confluence in the hypergeometric equation.

Theorem 18 (Parise, Lambert, Rousseau). Let ∆(x, ) be a non-resonant system (25) with A(x, ) = Λ(x, ) + (x 2 -)R(x, ) for some analytic germ R(x, ), and let ∆(x, ) be its formal normal form [START_REF] Zhang | Confluence et phénomène de Stokes[END_REF]. Then there exists a family of ramified "spiraling" domains X( √ ), √ ∈ S, as in Theorem 9 (i) (Figure 3) on which there exists a normalizing transformation T (x, ) = O(x 2 -) has zeros on the diagonal. We search for U (x, √ ), such that y

D = I + U (x, √ ) -1 y satisfies (x 2 -) dy D dx -Λ(x, ) + (x 2 -)D(x, √
) y D = 0, 5 Zhang also unfolds the Laplace integral (3), unlike us he chooses to unfold the kernel e -ξ x dξ by

x- √ ξ x+ √ ξ
for some diagonal matrix D(x, √ ), and set

T D (x, √ ) = e x √ D(s, √
) ds .

The matrix U (x, √ ) is solution to

(x 2 -) dU dx = ΛU -U Λ + (x 2 -) R (I + U ) -(I + U )D ,
where one must set D to be equal to the diagonal of R (I + U ). Therefore the off-diagonal terms of U = (u ij ) n i,j=1 are solution to the system of n(n -1) equations

(x 2 -) du ij dx = (λ i -λ j )u ij + (x 2 -) r ij + k =j r ik u kj -u ij r jj -u ij k =j r jk u kj ,
and one can apply Theorem 9.

2.4 Remark on generalization to singularities of greater multiplicities.

Saddle-node singularities of codimension k (multiplicity k + 1) unfold generically as

(x k+1 + k-1 x k-1 + . . . + 0 ) dy dx = M y + f (x, , y), (x, , y) ∈ C × C k × C m . ( 28
)
The case of dimension m = 1 was studied in [START_REF] Rousseau | Analytical moduli for unfoldings of saddle-node vector filds[END_REF]; their construction of the centermanifold should probably generalize also to the case m ≥ 1 with M having spectrum of Poincaré type. The non-Poincaré situation is hinted in [START_REF] Hurtubise | Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank k[END_REF] where a generalization of Theorem 18 on sectoral normalization of unfolded irregular singularities of linear systems is given. As in Remark 8, the domains constructed in [START_REF] Hurtubise | Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank k[END_REF] are linked to the real phase space of the complex vector fields e i( π 2 -α) (x k+1 + k-1 x k-1 + . . . + 0 ) ∂ ∂x (cf. [START_REF] Branner | Classification of complex polynomial vector fields in one complex variable[END_REF]). Theorem 15 on summability of the unique formal power series solution in (x, ) seems for some reason to be rather particular to the codimension k = 1 (multiplicity 2). Already in the case of (28) with the derivation on the left side (x k+1 -0 ) ∂ ∂x , the sectors in 0 -space for the domains constructed in [START_REF] Hurtubise | Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank k[END_REF] are only of opening > 2π k , while one would rather want them > (k+1)π k in order to correspond with the expected Gevrey order ( 1 k , k+1 k ) of the formal solution in (x, 0 ). Let us note however, that the special case of (28) with the derivation x(x k -1 ) ∂ ∂x , which is invariant with respect to the rotation x → e 2πi k x, can be reduced to x(x -1 ) ∂ ∂ x by the ramification x = x k and a rank reduction (cf. [START_REF] Canalis-Durand | Monomial summability and doubly singular differential equations[END_REF]), therefore it does not encounter the above mentioned issue.

Preliminaries on Fourier-Laplace transformations

We will recall some basic elements of the classical theory of Fourier-Laplace transformations on a line in the complex plane. The book [START_REF] Doetsch | Introduction to the Theory and Application of the Laplace Transformation[END_REF] can serve as a good reference.

For α ∈ R and a locally integrable function φ : e iα R → C, one defines its two-sided Laplace transform

L α [φ](t) = +∞e iα -∞e iα φ(ξ) e -tξ dξ (29)
whenever it exists. Later on, in Section 4, we will replace the variable t by the time variable t(x, ) (9) of the vector field [START_REF] Costin | On Borel summation and Stokes phenomena for rank-1 nonlinear systems of ordinary differential equations[END_REF]. , that vanishes at infinity uniformly in each closed substrip

T A 1 ,B 1 α ⊆ T A,B α , by B α [f ](ξ) = 1 2πi V.P. C+∞ie -iα C-∞ie -iα f (t) e tξ dt, for ξ ∈ e iα R, (30)
where V.P.

C+∞ie -iα

C-∞ie -iα stands for the "Cauchy principal value" lim

N →+∞ C+N ie -iα C-N ie -iα and C ∈ T A,B α .
The two-sided Laplace transformation (29) and the Borel transformation (30) of analytic functions are inverse one to the other when defined. We will only need the following particular statement.

|φ| A 1 ,B 1 e iα R ≤ 1 2π sup A 1 ≤C≤B 1 τ ∈C+R |f (ie -iα τ )| dτ for A < A 1 < B 1 < B, and
f (t) = L α [φ](t) for all t ∈ T A,B α .
2) Let f be as in 1) with B = B 1 = +∞, the strips being replaced by half-planes.

Then the Borel transform φ(ξ) = B α [f ](ξ) is absolutely convergent and continuous on e iα R, and φ(ξ) = 0 for ξ ∈ ]-∞e iα , 0[,

|φ| A 1 ,+∞ e iα R := sup s∈R + |φ(e iα s) e -A 1 s | ≤ 1 2π sup C≥A 1 τ ∈C+R |f (ie -iα τ )| dτ, and
f (t) = L α [φ ](t) = +∞e iα 0 φ(ξ) e -tξ dξ
is the one-sided Laplace transform of φ in the direction α.

Proof. See [START_REF] Doetsch | Introduction to the Theory and Application of the Laplace Transformation[END_REF], Theorems 28.1 and 28.2.

Under the assumptions of Theorem 24, the Borel transformation converts derivative to multiplication by -ξ:

B α [ df dt ](ξ) = -ξ • B α [f ](ξ),
which can be seen by integration by parts. It also converts the product to the convolution:

B α [f •g] (ξ) = [B α [f ] * B α [g]] α (ξ), defined by φ * ψ α (ξ) = ψ * φ α (ξ) := +∞e iα -∞e iα φ(ζ) ψ(ξ -ζ) dζ. (31)
Indeed, we have

L α [φ * ψ](t) = L α [φ](t) • L α [ψ](t)
using Fubini theorem and Theorem 24, and the assertion is obtained by the inversion theorem of the Laplace transform:

B α L α [φ] (ξ) = 1 2 lim ν→0+ φ(ξ + e iα ν) + φ(ξ -e iα ν) (cf.
[8], Theorem 24.3), using the continuity of [φ * ψ] α (ξ).

Lemma 25 (Young's inequality).

|φ * ψ| A,B e iα R ≤ |φ| A,B e iα R • ψ A,B e iα R ( and ≤ φ A,B e iα R • |ψ| A,B e iα R ), φ * ψ A,B e iα R ≤ φ A,B e iα R • ψ A,B e iα R .
Proof. Observe that

(|e -As | + |e -Bs |) ≤ (|e -Aσ | + |e -Bσ |) • (|e -A(s-σ) | + |e -B(s-σ) |), (32)
the rest follows easily.

Remark 26 (Convolution of analytic functions on open strips

). In the subsequent text, rather then dealing with functions on a single line e iα R, one will work with functions which are analytic on some open strips in the ξ-plane (also called the Borel plane), or on more general regions obtained as connected unions of open strips of varying directions α.

If Ω j , j = 1, 2, are two open strips of the same direction α, and φ j ∈ O(Ω j ) are two analytic functions of bounded • A,B c j +e iα R -norms, then their convolution

[φ 1 * φ 2 ](ξ) = c 1 +∞e iα c 1 -∞e iα φ 1 (ζ) φ 2 (ξ -ζ) dζ, ξ ∈ c 1 + c 2 + e iα R, c j ∈ Ω j
is well defined and analytic on the strip Ω 1 + Ω 2 .

Definition 27 (Dirac distributions in the Borel plane). It will be convenient to introduce for each a ∈ C the Dirac mass distribution δ a (ξ), acting on the Borel plane as shift operators ξ → ξ -a: If φ(ξ) is an analytic function on some strip Ω in a direction α one defines δ a * φ (ξ) := φ(ξ -a), its translation to the strip Ω -a. With this definition, the operator δ 0 plays the role of the unity of convolution. One can represent each δ a as a "boundary value" of the function

1 2πi (ξ-a) (cf. [5]): Let δ ↓ a (ξ) := 1 2πi (ξ-a) C [a,a+∞ ie iα ) , δ ↑ a (ξ) := 1 2πi (ξ-a) C [a,a-∞ ie iα ) ,
be its restrictions to the two cut regions (see Figure 8), one then writes

δ a (ξ) = δ ↓ a (ξ) -δ ↑ a (ξ),
and defines the convolution and the Laplace transform involving δ a by integrating each term δ ↓ a (resp. δ ↑ a ) along deformed paths γ ↓ α (resp. γ ↑ α ) of direction α in their respective domains as in Figure 8, 4

δ a * φ (ξ) = V.P. γ ↓ α -γ ↑ α 1 2πi (ζ-a) φ(ξ -ζ) dζ = φ(ξ -a), L α [δ a ](t) = V.P. γ ↓ α -γ ↑ α 1 2πi (ξ-a) e -tξ dξ = e -at .
The unfolded Borel and Laplace transformations associated to the vector field (x 2 -) ∂ ∂x

In this section we define the unfolded Borel and Laplace transformations B α , L α (8) and summarize their basic properties. We need to specify:

-the branch of the multivalued time function t(x, ) (9) of the kernel, -the paths of integration, -the domains in x-space and ξ-space where the transformations live, -sufficient conditions on functions for which the transformations exist. We provide these depending analytically on a root parameter

√ ∈ C.
Here √ is to be interpreted simply as a symbol for a new parameter (a coordinate on the " √ -plane"), that naturally projects on the original parameter = ( √ ) 2 .

Let t(x, ) (9) be the complex time of the vector field -(x 2 -) ∂ ∂x with t(∞, ) = 0, which is well defined for

x ∈ CP 1 [- √ , √
] and extended analytically as a ramified function. Let us remark that the limit of the Riemann surface of t(•, ) as → 0 is composed of Z-many complex planes identified at the origin, but the Riemann surface of t(•, 0) is just the punctured x-plane in the middle.

Definition 28. For 0 ≤ Λ < π 2 √ | | , denote X(Λ, √ ) := {x ∈ C | |t(x, ) -k πi √ | > Λ, k ∈ Z}
an open neighborhood of the origin in the x-plane (of radius ∼ 1 Λ when is small) containing the roots ± √ .

For a direction α ∈ R and 0

≤ Λ < -1 2 e iα πi √ , let T ± α (Λ,

√

) be as in ( 16), a slanted strip in direction -α + π 2 in the t-coordinate passing between two discs of radius Λ centered at 0 and ∓ πi √ , and let X± α (Λ, √ ) be its projection to the xplane (see Figures 9 and10 ) projected to the x-plane for α = π 2 . The integration paths γ ± α are projections of the paths c ± -ie -iα R in the t-coordinate (they have opposite direction than those in Figure 9).

In order to apply the Borel transformation (30) in a direction α to a function f analytic on the neighborhood X(Λ, √ ), one may choose to lift f to the Riemann surface of t(

B ± α [f ](ξ, √ ) = 1 2πi c ± +∞ie -iα c ± -∞ie -iα f (x(t, )) e tξ dt, c ± ∈ T ± α (Λ,

√

).

For √ = 0 : If x ∈ X± α (Λ, √ ), then t(x, ) = -1 2 √ log √ -x √ +x ± πi , B ± α [f ](ξ, √ ) = e ∓ ξπi 2 √ • 1 2πi γ ± α f (x) x 2 - √ -x √ +x -ξ 2 √ dx, (33)
where the integration path γ ± α (see Figure 10) follows a real time trajectory of the vector field ie

-iα (x 2 -) ∂ ∂x inside X± α (Λ, √ ). Hence B - α [f ](ξ, √ ) = e ξπi √ • B + α [f ](ξ, √ ) (34) = -B + α+π [f ](ξ, e πi √ ), (35)
as Xα (Λ,

√ ) = X+ α+π (Λ, e πi √ ).
For √ = 0 :

B ± α [f ](ξ, 0) = 1 2πi γ ± α f (x) x 2 e ξ x dx, (36)
where γ ± α is a real time trajectory of the vector field ie -iα x 2 ∂ ∂x inside X± α (Λ, 0). It is the radial limit of the precedent case as √ → 0,

B ± α [f ](ξ, 0) = lim ν→0+ B ± α [f ](ξ, ν √ ).
The transformation B + α [f ](ξ, 0) is the standard analytic Borel transform (5) in direction α, and

B - α [f ](ξ, 0) = -B + α+π [f ](ξ, 0). ( 37
)
The following proposition summarizes some basic proprieties of these unfolded Borel transformations.

Proposition 30. Let α be a direction, and suppose that arg

√ ∈ ]α -π, α[ if = 0. 1) If √ = 0, let a function f ∈ O( X± α (Λ, √ )), be uniformly O(|x- √ | a |x+ √ | b ) at the points ± √ , for some a, b ∈ R with a + b > 0. Then the transforms B ± α [f ](ξ,

√

) converge absolutely for ξ in the strip

Ω α = {-(e -iα 2b √ ) > (e -iα ξ) > (e -iα 2a √ )}, see Figure 11, (38)
± α [f ]| Λ 1 ,λα-Λ 1 c+e iα R on any line c + e iα R ⊆ Ω α .
2) If √ = 0 and a + b > 0, then for ξ ∈ Ω α (defined in (38))

B + α [(x- √ ) a (x+ √ ) b ](ξ, √ ) = -ξπi 2 √ +aπi • (2 √ ) a+b-1 • 1 2πi B(a -ξ 2 √ , b + ξ 2 √ ),
where B is the Beta function.

3) In particular, for a positive integer n, and ξ in the strip in between 0 and 2n √ ,

B ± α [(x- √ ) n ](ξ, √ ) = χ ± α (ξ, √ ) • ξ (n-1) -2 √ • ξ (n-2) -2 √ • . . . • ξ 1 -2 √ ,
where for

√ = 0 and α ∈ ] arg √ , arg √ + π[ χ + α (ξ, √ ) := 1 1 -e ξπi √ , χ - α (ξ, √ ) := -1 1 -e -ξπi √ = χ + α (ξ, √ ) -1, (39)
and for

√ = 0 χ + α (ξ, 0) :=      1, if ξ ∈ ]0, +∞e iα [, 1 2 , if ξ = 0, 0, if ξ ∈ ]-∞e iα , 0[, χ - α (ξ, 0) := χ + α (ξ, 0) -1. Let us remark that χ ± α (ξ, ν √ ) ν→0+ ----→ χ ± α (ξ, 0) for ξ ∈ e iα R {0}. 4) If f (x) is analytic on an open disc of radius r > 2 | | centered at x 0 = - √ (or x 0 = √ ) and f (x 0 ) = 0, then B ± α [f ](ξ, √ ) = χ ± α (ξ, √ ) • φ(ξ)
where φ is is an entire function with at most exponential growth at infinity ≤

e |ξ| R-2 √ | | • O( |ξ|) for any 2 | | < R < r (where the big O is uniform for (ξ, √ ) → (∞, 0)).

5) For

√ = 0, c ∈ C, the Borel Transform B ± α x- √ x+ √ c (ξ, √ ) = δ 2c √ (ξ) is
the Dirac mass at 2c √ , acting as translation operator on the Borel plane by ξ → ξ -2c √ :

B ± α x- √ x+ √ c • f (ξ, √ ) = B ± α [f ](ξ -2c √ , √ ).
Remark 31. Although in 1) and 2) of Proposition 30 the function

f = O((x - √ ) a (x + √ ) b
), a + b > 0, might not vanish at both points ± √ as demanded in Definition 29, one can write

(x- √ ) a (x+ √ ) b = x- √ x+ √ c (x- √ ) a-c (x+ √ ) b+c , for any -b < c < a,
hence, using 5) of Proposition 30, the Borel transform B ± α [f ] is well defined as the translation by 2c √ of the Borel transform of the function f

• x- √ x+ √ -c
, this time vanishing at both points:

B ± α [f ](ξ, √ ) = B ± α f • x- √ x+ √ -c (ξ -2c √ , √ ).
Proof of Proposition 30. 1) For √ = 0, one can express

x- √ = 2 √ e -2 √ t 1 -e -2 √ t , x+ √ = 2 √ 1 1 -e -2 √ t .
If ξ is in the strip Ω α , ξ ∈ 2c √ + e iα R for some c ∈ ]-b, a[, one writes

B ± α [(x- √ ) a (x+ √ ) b ](ξ, √ ) = = (2 √ ) a+b 1 2πi C ± +e -iα iR (e -2 √ t ) a-c (1 -e -2 √ t ) a+b e (ξ-2c √ )t dt.
The term e (ξ-2c √ )t stays bounded along the integration path, while the term e -2 √ t (a-c)

(1-e -2 √ t ) a+b decreases exponentially fast as t -C ± → +∞ie -iα and t -C ± → -∞ie -iα , if α / ∈ arg √ + πZ.
2) From (33)

B + α [(x- √ ) a (x+ √ ) b ](ξ, √ ) = = -e -ξπi 2
√ +aπi 1 2πi γ + α ( √ -x) a-1-ξ 2 √ ( √ +x) b-1+ ξ 2 √ dx = e -ξπi 2 √ +aπi (2 √ ) a+b-1 1 2πi 1 0 (1 -s) a-1-ξ 2 √ s b-1+ ξ 2 √ ds, substituting s = √ +x 2 √ . For α = arg √ + π
2 , the integration path γ + α (= a real trajectory of the vector field e -i arg √ (x 2 -) ∂ ∂x ) can be chosen as the straight oriented segment ( √ , -√ ). The result follows.

3) From 2) using standard formulas. 4) For x 0 = -√ , one can write f (x) as a convergent series f (x) = +∞ n=1 a n (x+ √

) n with |a n | ≤ CK n for some C > 0 and

1 r < K < 1 R . Hence 1 -e ξπi √ • B + [f ](ξ, √ ) = +∞ n=1 a n ξ n-1 -2 √ • • • ξ 1 -2 √ =: +∞ n=1 b n (ξ, √ ),
where the series on the right is absolutely convergent for any ξ ∈ C. Indeed, let N = N (ξ, √ ) be the positive integer such that

|ξ| N +1 ≤ R -2 | | < |ξ| N , (40)
then

• for n ≥ N + 1: K • ( |ξ| n + 2 | |) ≤ RK, • for n ≤ N : 2 | | < 2 √ | | R-2 √ | | |ξ| n and hence K • ( |ξ| n + 2 | |) ≤ K|ξ| n 1 + 2 √ | | R-2 √ | | ≤ 1 n • |ξ| R-2 √ | | . +∞ n=1 |b n (ξ, √ )| = N -1 n=0 |b n+1 (ξ, √ )| + +∞ n=N |b n+1 (ξ, √ )| ≤ N -1 n=0 CK 1 n! |ξ| R-2 √ | | n + CK 1 N ! |ξ| R-2 √ | | N • +∞ n=N (RK) n-N ≤ CKe |ξ| R-2 √ | | + CK • Γ |ξ| R-2 √ | | -1 |ξ| R-2 √ | | |ξ| R-2 √ | | • 1 1-RK = e |ξ| R-2 √ | | • CK + CK 1-RK |ξ| 2π(R-2 √ | |) + O R-2 √ | | |ξ| , using (40) and the Stirling formula: Γ(z) -1 = e z z • z 2π + O 1 √ z , z → +∞.

5) From the definition.

There is also a converse statement to point 1) of Proposition 30.

Proposition 32. Let = 0 and α ∈ ] arg √ , arg √ + π[. If φ(ξ) is an analytic function in a strip Ω α (38), with a+b > 0, such that it has a finite norm |φ| Λ,λα-Λ 2c √ +e iα R on each line 2c √ + e iα R ⊆ Ω α , for some 0 ≤ Λ < λ α ( √ ) := -e iα πi 2 √ ), then the unfolded Laplace transform of φ L α [φ](x, √ ) = 2c √ +∞e iα 2c √ -∞e iα φ(ξ)e -t(x, )ξ dξ, c ∈ ]-b, a[ (41)
is analytic on the domain

X+ α (Λ, √ ), and is uniformly o(|x- √ | a 1 |x+ √ | b 1 ) for any a 1 < a, b 1 < b, on any sub-domain X+ α (Λ 1 , √ ), Λ 1 > Λ.
Proof. This is a reformulation of Corollary 22, which also implies that L α [φ] is

o x- √ x+ √ c
for any -b < c < a.

Definition 33 (Borel transform of x). We know form Proposition 30 that for

√ = 0, B ± α [x + √ ] = χ ± α in the strip in between -2 √ and 0, while B ± α [x - √ ] = χ ±
α in the strip in between 0 and 2 √ , and the function χ ± α has a simple pole at 0 with residue Res 0 χ ± α = √ πi , therefore

B ± α [x + √ ] -B ± α [x - √ ] = 2 √ δ 0
in the sense of distributions (see Section 27), where δ 0 is the Dirac distribution (identity of convolution). Hence one can define the distribution

B ± α [x] := B ± α [x - √ ] + √ δ 0 = B ± α [x + √ ] - √ δ 0 .
Correspondingly, the convolution of B ± α [x] with a function φ, analytic on an open strip in direction α, is then defined as Figure 12: The integration paths c

[B ± α [x] * φ] α (ξ, √ ) = c 1 +e iα R φ(ξ -ζ) χ ± α (ζ, √ ) dζ + √ φ(ξ), c 1 ∈ ]0, 2 √ [ = c 2 +e iα R φ(ξ -ζ) χ ± α (ζ, √ ) dζ - √ φ(ξ), c 2 ∈ ]-2 √ , 0[.

Remark on Fourier expansions

• + i √ R (• = L, R) in the time t-coordinate.
If f is analytic on a neighborhood of x = √ (resp. x = -√ ), then the lifting of f to the time coordinate, f (x(t, )), is πi √ -periodic in the half-plane (e i arg √ t) > Λ (resp. (e i arg √ t) < -Λ ) for Λ large enough, and can be written as a sum of its Fourier series expansion:

f (x) = +∞ n=0 a R n e -2n √ t(x) = +∞ n=0 a R n • x- √ x+ √ n , resp. f (x) = +∞ n=0 a L n e 2n √ t(x) = +∞ n=0 a L n • x+ √ x- √ n .
The Borel transform (30

) of f (x(t, )) on the line c R + i √ R (resp. c L + i √ R) is equal to the formal sum of distributions B R [f ](ξ, √ ) := 1 2πi c R + i √ ∞ c R -i √ ∞ f (x(t, )) e tξ dt = +∞ n=0 a R n δ 2n √ (ξ), resp. B L [f ](ξ, √ ) := 1 2πi c L + i √ ∞ c L -i √ ∞ f (x(t, )) e tξ dt = +∞ n=0 a L n δ -2n √ (ξ).
These transformations were studied by Sternin and Shatalov in [START_REF] Yu | On the Confluence phenomenon of Fuchsian equations[END_REF]. One can connect the coefficients a • n of these expansions to residues of the unfolded Borel transforms

B ± α , arg √ < α < arg √ + π, a R 0 = f ( √ ), a R n = 2πi Res 2n √ B ± α [f ], n ∈ N >0 , a L 0 = f (- √ ), a L n = 2πi Res -2n √ B ± α [f ], n ∈ N >0 , (the residues of B + α [f ] and B - α [f ] at the points ξ ∈ 2 √ Z are equal).
Remark 34. Without providing details, let us remark that one could follow [START_REF] Yu | On the Confluence phenomenon of Fuchsian equations[END_REF] and apply these Borel transformations B R (resp. B L ) to the system [START_REF] Hurtubise | Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank k[END_REF] to show the convergence of its unique local analytic solution at x = √ = 0 (resp. x = -√ = 0) to a Borel sum in direction arg √ of the formal solution ŷ0 (x) of the limit system, when √ → 0 radially in a sector not containing any eigenvalue of M , as stated in Theorem 12.

Solution to the equation (12) in the Borel plane

We will use the unfolded Borel transformation B ± α to transform the equation ( 12) :

(x 2 -) dy dx = M y + f (x, , y)
to a convolution equation in the Borel plane, and study its solutions there. We write the function f (11) as

f (x, , y) = |l|≥0 f l (x, ) y l , (42)
where

y l := y l 1 1 • . . . • y lm m for each multi-index l = (l 1 , . . . , l m ) ∈ N m , |l| = l 1 + . . . + l m , and f 0 (x, ) = O(x 2 -), f l (x, ) = O(x) + O( ), for |l| = 1. Let a vector variable υ = υ(ξ, √ ) correspond to the Borel transform B ± α [y](ξ, √ ), with α ∈ ] arg √ , arg √ + π[ if √ = 0.
Then the equation ( 12) is transformed to a convolution equation in the Borel plane

ξυ = M υ + |l|≥0 B ± α [f l ] * υ * l , (43)
where

υ * l := υ * l 1 1 * . . . * υ * lm m
is the convolution product of components of υ, each taken l i -times, the convolutions being done in the direction α,

B ± α [f l ](ξ,

√

) is a sum of an analytic function and a multiple of the Dirac distribution δ 0 (ξ). In Proposition 36, we will find a unique analytic solution υ ± (ξ, √ ) of the convolution equation (43) as a fixed point of the operator

G ± [υ](ξ, √ ) := ξI -M -1 |l|≥0 B ± [f l ] * υ * l (ξ, √ ) (44)
on a domain Ω( √ ) in the ξ-plane, obtained as union of strips Ω α ( √ ) of continuously varying direction α, passing in between the points -2 √ and 2 √ , that stay away from the eigenvalues of the matrix M (see Figure 13). In general, several ways of choosing such a domain Ω( √ ) are possible, depending on its position relative with respect to the eigenvalues of M . Different choices of the domain Ω( √ ) will, in general, lead to different solutions υ ± (x, √ ) of ( 43), as shown in Example 37 below.

Definition 35 (Family of regions Ω( √ ) in the Borel plane). Let the two directions β 1 < β 2 and an arbitrarily small angle 0 < η < 1 4 (β 2 -β 1 ) be as in Definition 7, and let ρ > 0 be small enough so that for | √ | < ρ none of the closed strips

Ω α ( √ ) = c∈[-3 2 √ , 3 2 √ ] c + e iα R, (45)
with α ∈ ]β 1 + η, β 2 -η[, contains any eigenvalue of M ± √ . We define a family of regions Ω( √ ) in the ξ-plane depending parametrically on √ ∈ S [START_REF] Malgrange | Sommation des séries divergentes[END_REF] as

Ω( √ ) := α Ω α ( √ ),
α as in [START_REF] Martinet | Problèmes de modules pour des équations différentielles non linéaires du premier ordre[END_REF], and denote Ω :

= √ ∈S Ω( √ ) ( 46
)
their union in the (ξ, √ )-space. Let ρ, η > 0 be as in the definition of S (15), and let 0 ≤ Λ < π sin η 2ρ . For a vector function φ = (φ 1 , . . . , φ m ) : Ω → C m , we say that it is analytic on Ω, if it is continuous on Ω, analytic on the interior of Ω, and φ(•, √ ) is analytic on Ω( √ ) for all √ ∈ S. We define the norms

|φ| Λ Ω := max i sup √ , α sup c∈Ωα( √ ) |φ i | Λ,λα-Λ c+e iα R , φ Λ Ω := max i sup √ , α sup c∈Ωα( √ ) φ i Λ,λα-Λ c+e iα R ,
|φ * ψ| Λ Ω ≤ min |φ| Λ Ω • ψ Λ Ω , φ Λ Ω • |ψ| Λ Ω (47) φ * ψ Λ Ω ≤ φ Λ Ω • ψ Λ Ω . (48)
We extend these relations also to the Dirac distribution with mass at 0 by setting

|δ 0 | Λ Ω = δ 0 Λ Ω = 1.
Proposition 36 (Solution to the convolution equation ( 43)). Suppose that the vector function f (x, , y) in the equation (12) are analytic for

x ∈ X(Λ 1 , √ ), m i=1 |y i | < 1 L 1 , | √ | < ρ 1 , for Λ 1 , L 1 , ρ 1 > 0.
Then there exists Λ > Λ 1 , 0 < ρ ≤ ρ 1 , and a constant c > 0, such that the operator While for = 0 the germ of the solutions υ + (•, 0) at ξ = 0 equals to the Borel transform of the unique formal solution ŷ0 (x) = ŷ(x, 0) [START_REF] Lambert | Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank 1[END_REF], and therefore it is independent of the domain Ω(0), this is no longer true when one unfolds. The reason is that the convolution is no longer defined locally, but involves integration on a whole line c + e iα R. As the following example shows, the analytic solutions υ ± of (43) therefore depend in general on the position of the region Ω( √ ) with respect to the eigenvalues of M , and are not analytic extensions one of another. We'll see later (Corollary 44) that their difference is exponentially small in √ for each fixed small ξ.

G + : φ(ξ, √ ) → G + [φ](ξ, √ ) (44
Example 37. Let u satisfy

(x 2 -) du dx = u + (x 2 -), (49)
and let y = (x 2 -)u ; it satisfies

(x 2 -) dy dx = y + 2xy + (x 2 -) 2 . ( 50
)
The Borel transform of the equation ( 49) is

ξφ ± α = φ ± α + ξχ ± α , φ ± α = B ± α [u] therefore φ ± α (ξ, √ ) = ξ ξ-1 χ ± α (ξ,

√

), which is independent of the direction α. This is no longer true for the solution υ

± α = φ ± α * B ± α [x 2 -] of the Borel transform of the equation (50) ξυ ± α = υ ± α + 2B ± [x] * υ ± α + χ ± α • (ξ 3 -4 ξ). If, for instance, ( √ ) < 0, and arg √ < α 1 < 0 < α 2 < arg √ + π, then the strips Ω α 1 ( √ ), Ω α 2 ( √ ) (45) in directions α 1 , α 2
, are separated by the point ξ = 1, and one easily calculates that for ξ

∈ Ω α 1 ( √ ) ∩ Ω α 2 ( √ ) υ ± α 1 (ξ, √ ) -υ ± α 2 (ξ, √ ) = (ξ -1) χ ± α (1, √ ) χ ± α (ξ -1, √ ),
i.e. the two solutions υ ± α 1 , υ ± α 2 differ near ξ = 0 by a term that is exponentially flat in √ .

To prove Proposition 36 we will make use of the following technical lemmas which will allow us to estimate the norms of

G + [φ]. Lemma 38. There exists a constant C = C(Λ 1 , η) > 0 such that, if h ∈ O(X(Λ 1 , √ )), | √ | < ρ, and Λ 1 < Λ < π sin η 2ρ
(where η, ρ > 0 are as in [START_REF] Martinet | Problèmes de modules pour des équations différentielles non linéaires du premier ordre[END_REF], ( 15)), then

|B + α [(x 2 -)h]| Λ Ω ≤ Cρ sup x∈X(Λ 1 , √ ) |h(x)|.
Proof. Essentially, one needs to estimate the integral (e iα t)=Λ

x-

√ x+ √ c d|x|, with c ∈ [-3 4 , 3 4 ] and α ∈ ] arg √ + η, arg √ + π -η[.
Lemma 39. Let φ be an analytic function on Ω with a finite |φ| Λ Ω (resp. φ Λ Ω ). Then its convolution with the distribution B + [x] (Definition 33) is again an analytic function on Ω whose norm satisfies

|B + [x] * φ| Λ Ω ≤ |φ| Λ Ω • ρ + χ + α Λ Ω L , (51)
resp.

B + [x] * φ Λ Ω ≤ φ Λ Ω • ρ + χ + α Λ Ω L , (52)
where χ + α is given in (39), ρ is the radius of S, and

Ω L ( √ ) = Ω( √ ) ∩ Ω( √ )-2 √ , for each √ ∈ S. (53)
Proof. It follows from Definition 33 and 2 √ -periodicity of χ + α .

Lemma 40. If φ, ψ : Ω → C m are analytic vector functions such that φ Λ Ω , ψ Λ Ω ≤ a, then for any multi-index l ∈ N m , |l| ≥ 1,

|φ * l -ψ * l | Λ Ω ≤ |l| • a |l|-1 • |φ -ψ| Λ Ω .
The same holds for the • Λ Ω -norm as well. Proof. Writing φ * l = φ i 1 * . . . * φ i |l| , i j ∈ {1, . . . , m}, we have

φ * l -ψ * l = (φ i 1 -ψ i 1 ) * φ i 2 * . . . * φ i |l| + ψ i 1 * (φ i 2 -ψ i 2 ) * φ i 3 * . . . * φ i |l| + . . . + ψ i 1 * . . . * ψ i |l|-1 * (φ i |l| -ψ i |l| ).
The statement now follows from the convolution inequalities (47) (resp. (48)).

B + [f 0 ] Λ Ω ≤ ρK 0 , B + [f l ] Λ Ω ≤ ρK 1 L, if |l| = 1, B + [f l ] Λ Ω ≤ K 2 |l| l L |l| , |l| ≥ 2,
for some K 0 , K 1 , K 2 > 0. Moreover, if we take Λ sufficiently large and ρ sufficiently small, then we can make the constant K 0 arbitrarily small. Let

σ = max (ξ, √ )∈Ω (Iξ -M ) -1 • 1 . . . 1 ,
then σ < +∞ if the radius ρ of S is small, and suppose that ρK 0 is small enough, so that there is c > 0 satisfying (54) and ( 55

) below. Then if φ Λ Ω ≤ c G + [φ] Λ Ω ≤ σ • ρK 0 + ρK 1 cmL + K 2 +∞ k=2 (cmL) k ≤ c, (54)
using (48), and similarly,

|G + [φ]| Λ Ω ≤ max{c, |φ| Λ Ω } if φ Λ Ω ≤ c. And if φ Λ Ω , ψ Λ Ω ≤ c, then |G + [φ] -G + [ψ]| Λ Ω |φ -ψ| Λ Ω ≤ σ • ρK 1 mL + K 2 mL +∞ k=2 k(cmL) k-1 ≤ 1 2 , (55)
using Lemma 40 and the convolution inequality (47). The same holds for the

• Λ Ω -norm. Hence the operator G + is | • | Λ Ω -contractive, and the sequence G + n [0] converges, as n → +∞, |•| Λ Ω -uniformly to an analytic function υ + satisfying G + υ + ] = υ + . From (34) it follows that B -[f l ] = e ξπi √ • B + [f l ], hence G -[υ -] = G -[e ξπi √ υ + ] = e ξπi √ • G + [υ + ] = e ξπi √ • υ + = υ -is a fixed point of G -.
Proposition 41 (Poincaré case). If the spectrum of M is of Poincaré type, i.e. if it is contained in a sector of opening < π, then, for small √ , the region Ω( √ ) may be chosen so that it has all the eigenvalues of M on the same side-let's say the side where 2 √ is. In such case, let Ω 1 ( √ ) be the extension of Ω( √ ) to the whole region on the opposite side (see Figure 14). The solutions υ ± (ξ, √ ) of Proposition 36 can be analytically extended to Ω 1 ( √ ) (-2 √ )N >0 with at most simple poles at the points -2 √ N >0 . The function υ ± χ ± α is analytic in Ω 1 and has at most exponential growth < Ce Λ|ξ| for some Λ, C > 0 independent of √ .

Proof. As in the proof of Proposition 36, the solution υ + is constructed as a limit of the iterative sequence of functions (G + ) n [0], n → +∞. We will show by induction that for each n, the function (G + ) n [0] is analytic on Ω 1 {ξ ∈ -2 √ N >0 } and has at most simple poles at the points ξ ∈ -2 √ N >0 , and that the sequence converges uniformly to υ + with respect to the norm

φ Λ Ω 1 := sup (ξ, √ )∈Ω 1 | φ χ + (ξ, √ )| e -Λ|ξ| . (56)
To do so we will introduce another norm || • Λ Ω 1 , defined in (60) below, such that the two norms satisfy convolution inequalities similar to those satisfied by | • | Λ Ω and • Λ Ω (Lemma 42 below). Then one can simply replicate the proof of Proposition 36 with the norm

• Λ Ω 1 in place of | • | Λ Ω and the norm || • Λ Ω 1 in place of • Λ Ω .
Figure 14: The extended regions Ω 1 ( √ ) in the Borel plane, together with the modified integration path Γ of the Laplace transform (compare with Figure 13). The limit region Ω 1 (0) := Suppose now that φ, ψ have at most simple poles at the points -2 express the convolution as

√ ∈S ν→0+ Ω 1 (ν √ ) (- 2
√ N >0 . If ξ is in Ω( √ ) ∪ 2Ω L ( √ ) (Ω L is defined in (53)), then Γ ξ = c + e iαR for some c ∈ [-3 2 √ , -1 2 √ ] ⊂ Ω L ( √ ). Else ξ ∈ 2Ω L ( √ ) -2k √ for some k ∈ N >0 ,

and one can

(φ * ψ)(ξ) = c-2k √ +e iα R φ(s) ψ(ξ -s) ds + 2πi k j=1 Res -2j √ φ • ψ(ξ + 2j √ ) = c+e iα R φ(t -2k √ ) ψ(ξ 0 -t) dt -2 √ k-1 j=0 φ χ + (-2(k -j) √ ) • ψ(ξ 0 -2j √ ), ( 57
) where c ∈ [-3 2 √ , -1 2 √ ] ⊂ Ω L ( √ ) and ξ 0 = ξ + 2k √ ∈ c + Ω L ( √ ), i.e. ξ -s ∈ Ω L ( √ )
, see Figure 15. We will use this formula to obtain an estimate for the norm

φ * ψ Λ Ω 1 , Λ ≥ 0. Since | 1 χ + (ξ, √ ) | ≤ (1 + |e sπi √ |)(1 + |e (ξ-s)πi √ |), cf. (32), we have | φ * ψ χ + (ξ)| e -Λ|ξ| ≤ sup s∈Ω L ( √ )-2k √ |φ(s)|(1 + |e sπi √ |) e -Λ|s| • ψ Λ Ω( √ ) + 2 | | • φ Λ Ω 1 • k-1 j=0 | ψ χ + (ξ 0 -2j √ )| e -Λ|ξ 0 -2j √ | , (58)
due to the 2 √ -periodicity of χ + . Let µ ≥ 1 be such that

1 + |e sπi √ | ≤ µ | 1 χ + (s, √ ) | for all s ∈ Ω L ( √ ) (59)
and define

| ψ Λ Ω 1 := µ ψ Λ Ω + sup √ ∈S ξ∈2Ω L ( √ ) 2 | | +∞ k=0 | ψ χ + (ξ -2k √ )| e -Λ|ξ-2k √ | . (60)
Then (58) implies that

φ * ψ Λ Ω 1 ≤ φ Λ Ω 1 • | ψ Λ Ω 1 .
Note that by 4) of Proposition 30, if f (x, )

x 2 -is analytic on {|x 2 -| < r 2 }×{| | < ρ 2 } for some r > 2ρ > 0, then for any Λ > 1 r-2ρ B α [f ] Λ Ω 1 < +∞, | B + α [f ] Λ Ω 1 <
√ ) = ψ χ + (0, √ ) = 0. Then φ * ψ Λ Ω 1 ≤ φ Λ Ω 1 • | ψ Λ Ω 1 , | φ * ψ Λ Ω 1 ≤ | φ Λ Ω 1 • | ψ Λ Ω 1 .
Proof. The first inequality is given in the proof of Lemma 41. We need to prove the second one. By definition

| φ * ψ Λ Ω 1 = µ φ * ψ Λ Ω + sup √ ∈S ξ∈2Ω L ( √ ) 2 | | +∞ k=0 | φ * ψ χ + (ξ -2k √ )| e -Λ|ξ-2k √ | .
The first term is smaller than

µ φ Λ Ω ψ Λ Ω ≤ µ 2 φ Λ Ω ψ Λ Ω since µ ≥ 1.
For the second term, using (57), (59) and 2 √ -periodicity of χ + , we have

+∞ k=0 | φ * ψ χ + (ξ -2k √ )| e -Λ|ξ-2k √ | ≤ c+e iα R µ +∞ k=0 | φ χ + (t-2k √ )| e -Λ|t-2k √ | • |ψ(ξ -t)| (1 + |e (ξ-t)πi √ |) e -Λ|ξ-t| d|t| + 2 | | +∞ k=0 k j=1 | φ χ + (- 2j
√ )| e -Λ|2j √ | • | ψ χ + (ξ -2(k-j) √ )| e -Λ|ξ-2(k-j) √ | ≤ sup ξ∈2Ω L ( √ ) +∞ k=0 | φ χ + (ξ -2k √ )| e -Λ|ξ-2k √ | • • µ ψ Λ Ω + sup ξ∈2Ω L ( √ ) 2 | | +∞ j=0 | ψ χ + (ξ - 2j
where α can vary as in [START_REF] Martinet | Problèmes de modules pour des équations différentielles non linéaires du premier ordre[END_REF], is a solution of (42) defined for t(x, ) in the domain

T ± ( √ ) = α T ± α (Λ, √
), (Figure 2). Both L[υ + ] and L[υ -] give the same ramified solution y(x, √ ) on a domain X( √ ) in the x-plane (Figure 3). ii) If the spectrum of M is of Poincaré type and υ ± (ξ, √ ) is defined on Ω 1 as in Proposition 41, with υ + Λ Ω 1 < +∞, then, for x ∈ X 1 ( √ ) ∩ { (e i arg √ t(x, )) < -Λ}, one may deform the integration path of the Laplace transform (61) to Γ, indicated in Figure 14, and use the Cauchy formula to express y ± (x, √ ), for √ = 0, as a sum of residues at the points ξ = -2k √ , k ∈ N >0 ,

L[υ ± ](x, √ ) = Γ υ ± (ξ, √ ) e -t(x, )ξ dξ = 2πi ∞ k=1 Res -2k √ υ ± • x+ √ x- √ k = -2 √ ∞ k=1 υ ± χ ± (-2k √ , √ ) • x+ √ x- √ k . ( 62
)
This series is convergent for x+ For = 0, the two solutions ῡ+ (•, 0) and υ-(•, 0) agree on the disc of radius R not containing any eigenvalue of M , and the two Laplace integrals can be compared directly. For = 0 this is no longer true. Instead, we will construct a set ῡ+ = φ 0 , φ 1 , . . . , φ N -1 , φ N = υ-of approximative solutions to (43) defined on some fixed neighborhoods U j of [-Re iα j , Re We can then estimate For any integrable bounded function φ : [-Re iα , Re iα ] → C we can still define its norms |φ| Λ e iα R and φ Λ e iα R by setting φ(ξ) = 0 outside of the interval, and hence use the same Youngs' inequalities for the convolution as before. One can then again prove that for each α there is a fixed point solution for the truncated version of the convolution operator G + α on the interval [-Re iα , Re iα ]. The following lemma implies that the difference between such solution of the truncated convolution equation and a true solution on e iα R, when the latter one exist, is uniformly bounded by K e Λ|ξ|+(2Λ-λα)(R-|ξ|) 1+|e ξπi √ | on the interval. One then extends analytically such the truncated solutions on small double-sectors around their interval of definition, so that the difference of each two of them φ j-1 , φ j with sufficiently close angles α j-1 , α j has the same kind of uniform bound.

Lemma 43. Suppose that is small enough so that λ α > 2Λ.

1 + |e ξπi √ | for ξ ∈ Ω 1 ( √ ) ∩ Ω 2 ( √ )
for some K > 0.

  : Throughout the text ]a, b[ (resp. [a, b]) denotes the open (resp. closed) oriented segment between two points a, b ∈ C; e iα R + = [0, +∞e iα [ is an oriented ray, and c + e iα R =]c -∞e iα , c + ∞e iα [, with α ∈ R, c ∈ C, is an oriented line.

Figure 1 :

 1 Figure 1: The rays λR + , λ ∈ Spec M divide the Borel plane in sectors. The integration path e iα R + of the Laplace transform L α [ B[ŷ 0 ]] varies in such sectors.

Figure 3 :

 3 Figure 3: Example of the spiraling domains X( √ ) of Theorem 9 (i) according to √ ∈ S.

Figure 4 :

 4 Figure 4: Example of the spiraling domains X 1 ( √ ) of Theorem 9 (ii) according to √ ∈ S 1 .

Figure 5 :

 5 Figure 5: The intersections of the domains T ± ( √ ˜ )∩T ∓ ( √ ¯ ) of Figure 2 for √ ¯ , √ ˜ = e πi √ ¯ (left) and their limits as → 0 (right).

Figure 6 :

 6 Figure 6: The spectrum of M in the Borel plane; the line e iα R is the dividing line of the Hadamard-Perron theorem and also the integration path of the Laplace transform L α (61).

2. 2 . 3

 23 Hadamard-Perron interpretation for = 0 and convergence of local analytic solutions.

Figure 7 :

 7 Figure 7: The resonant values of √ = λ 2n , λ ∈ Spec M √ , n ∈ N >0 , accumulate along the rays λR + , dividing the √ -plane in sectors on which the local analytic solutions near x = √ = 0 converge as √ → 0 to sectoral solutions. it follows from Proposition 10, that for √ ˜ in the intersection S ∩ e πi S and a fixed z, the difference |Y (z, √ ˜ ) -Y (z, -√ ˜ )| is exponentially flat in |˜ |, therefore by the Ramis-Sibuya theorem ([24], [1]) the bounded function Y (z, √ ) possesses an asymptotic expansion of Gevrey order 1 on S, equal to Ŷ (z, √) by its uniqueness, and therefore it is also its Borel sum on S (the opening of S is > π). We need yet to specify the domain of z on which this is true. First, remark that

∩ e πi 2 S< π 2 .

 22 [START_REF] Rousseau | Analytical moduli for unfoldings of saddle-node vector filds[END_REF] for some direction β covered by e -πi 2 S with |β -arg √ | Remark 14. The blow-up transforms[START_REF] Hurtubise | Complete system of analytic invariants for unfolded differential linear systems with an irregular singularity of Poincaré rank k[END_REF] to a singularly perturbed equation.

Definition 19 .- 1 D.Corollary 22 .Definition 23 .

 1912223 For A < B ∈ R, let us introduce the two following exponential weighted norms on locally integrable functions φ : e iα R → C:|φ| A,B e iα R = sup s∈R |φ(e iα s)| • |e -As | + |e -Bs | , φ A,B e iα R = s∈R |φ(e iα s)| • |e -As | + |e -Bs | ds.Proposition 20. If φ A,B e iα R < +∞, then the Laplace transform L α [φ](t) converges absolutely and is analytic for t in the closed stripT A,B α := {t ∈ C | A ≤ (e iα t) ≤ B}. Moreover, L α [φ](t) tends uniformly to 0 as t → ∞ in T ∞e iα φ(ξ) e -tξ dξconverges absolutely in the closed half-plane (e iα t) ≤ B, while the integral +∞e iα 0 φ(ξ) e -tξ dξ converges absolutely in the closed half-plane (e iα t) ≥ A. For the second statement see [8], Theorem 23.6. Lemma 21. If 0 < D < B-A 2 , then for any function φ : e iα R → C, φ A+D,B-D e iα R ≤ 4 D |φ| A,B e iα R . iα s)| |e -(A+D)s | + |e -(B-D)s | ds ≤ 0 -∞ |e Ds | ds • sup s∈R |φ(e iα s)| |e -As-2Ds | + |e -Bs | ≤ • 2|φ| A,B e iα R , since |e -As-2Ds | ≤ |e -Bs | ≤ |e -As | + |e -Bs |, for s < 0. The same kind of estimate is obtained also for +∞ 0 If |φ| A,B e iα R < +∞, then the Laplace transform L α [φ](t) converges absolutely and is analytic for t in the open strip T A,B α := {t ∈ C | A < (e iα t) < B}. Moreover, L α [φ](t) tends to 0 as t → ∞ uniformly in each T A 1 ,B 1 α ⊆ T A,B α . The Borel transformation is defined for any function f analytic on some open strip T A,B α

Theorem 24 . 1 )

 241 Let f ∈ O(T A,B α ) be absolutely integrable on each line C + ie -iα R ⊆ T A,B α and vanishing at infinity uniformly in each closed sub-strip of T A,B α . Then the Borel transform φ(ξ) = B α [f ](ξ) is absolutely convergent and continuous for all ξ ∈ e iα R,

Figure 8 :

 8 Figure 8: The domains of definition of δ ↓ a (resp. δ ↑ a ) together with the deformed integration paths γ ↓ α (resp. γ ↑ α ).

  ). More precisely, we shall consider them as subsets of the ramified Riemann surface of t(•, ). Then the limits of X± α (Λ,√) when → 0 radially, split each into two opposed discs of radius 1 2Λ tangent at the origin, of which only one lies inside the x-plane: X+ α (Λ, 0) is a disc centered at e iα 1 2Λ , and Xα (Λ, 0) is a disc centered at -e iα 1 2Λ (Figure10 (b)). The interior of the domains X ± ( √ ) of Theorem 9 are ramified unions of such domains X± α (Λ,√).

Figure 9 :Figure 10 :

 910 Figure 9: The domains T ± α (Λ, √ ) in the time coordinate t with the integration paths of the Borel transformation for α = π 2 .

Figure 11 :

 11 Figure 11: The strip Ω α in the ξ-plane.

For √ = 0 ,

 0 we have defined the Borel transformations B ± α for directions transverse to √ R: in fact, we have restricted ourselves to α ∈ ] arg √ , arg √ + π[. Let us now take a look at the direction arg √ . So instead of integrating on a line c ± + ie -iα R in the t-coordinate as in Figure 9, this time we shall consider an integrating path c R + i √ R in the half plane (e i arg √ t) > Λ (resp. c L + i √ R in the half plane (e i arg √ t) < -Λ ), see Figure 12.

Figure 13 :

 13 Figure 13: The regions Ω( √ ) and the eigenvalues λ 1 , . . . , λ m (here m = 3) of M in the ξ-plane according to √ ∈ S, together with integration paths e iα R of the unfolded Laplace transformation L α . where √ ∈ S and α as in (17), i.e. such that Ω α ( √ ) ⊂ Ω( √ ), and λ α ( √ ) = -e iα πi √ . Then the convolution of two analytic functions φ, ψ on Ω( √ ) does not depend on the direction α (17), and the norms |φ * ψ| Λ Ω , φ * ψ Λ Ω satisfy the Young's inequalities (Lemma 25):

  ) is well-defined and contractive on the space{φ : Ω → C m | φ is analytic on Ω, φ Λ Ω ≤ c, |φ| Λ Ω < +∞}with respect to both the • Λ Ω -norm and the | • | Λ Ω -norm. Hence the equation G + [υ + ] = υ + possesses a unique analytic solution υ + (ξ, √ ) on Ω, satisfying υ + Λ Ω ≤ c and |υ + | Λ Ω < +∞. Similarly, the vector function υ -(ξ, √ ) := e ξπi √ • υ + (ξ, √ ) is a unique analytic solution of the equation G -[υ -] = υ -on Ω.

  Proof of Proposition 36.If L > m • L 1 , then there exists K > 0 such that for each multi-index l ∈ N m , |f l (x, )| ≤ K • |l| l L |l| ,where for y ∈ C m , |y| = m i=1 |y i |, and where |l| l are the multinomial coefficients given by (y 1 + . . . + y m ) k = |l|=k |l| l y l , satisfying |l|=k |l| l = m k . It follows from Lemma 38, Lemma 39 and Lemma 21, that if Λ > Λ 1 , then

3 2 √ , 3 2 √ ] and [ξ -3 2 √ , ξ + 3 2 √ ], as in Figure 15 .

 3222215 √ )N >0 is composed of two sectors connected at the origin; the solution υ + (ξ, 0) vanishes on the lower sector, while the solution υ -(ξ, 0) vanishes on the upper one.Let us first show that if φ, ψ are two functions analytic on Ω1 ( √ ) (-2 √ N >0 ), then so is their convolution φ * ψ. If ξ ∈ Ω 1 ( √ ) √ R,then the analytic continuation of φ * ψ at the point ξ is given by the integral(φ * ψ)(ξ) = Γ ξ φ(s) ψ(ξ -s) dswith Γ ξ a symmetric path with respect to the point ξ 2 passing through the segments [-Note that when ξ approaches a point on ]-∞ √ , from one side or another, the values of the two integrals are identical, since both paths Γ ξ pass in between the same singularities.

Figure 15 :

 15 Figure 15: The integration path Γ ξ of convolution (φ * ψ)(ξ), ξ ∈ Ω 1 ( √ ).

  Let υ ± (ξ, √) be the solution of the convolution equation (43) on Ω, provided by Proposition 36, with bounded • Λ Ω -norm. Its Laplace transformL[υ ± ](x, √ ) = +∞e iα -∞e iα υ ± (ξ, √) e -t(x, )ξ dξ,

2 √

 2 | |Λ , and its coefficients are the same in both cases υ + and υ -. It defines a solution y 1 (x, √ ) of (12) on a domain X 1 ( √ ), analytic at x = -√ and ramified at x = √ (Figure4).

5.0. 2 e

 2 Proof of Proposition 10.Note first that for any integrable functionφ : e iα R → C with bounded | • | Λe iα R -norm, the difference between the two-sided Laplace transform L α [φ](t) and its truncation of the corresponding integral to [-Re iα , Re iα ] can be estimated, for t in the strip of convergenceT + α (Λ, √ ) = {Λ < (e iα t) < -( e iα πi √ ) -Λ}, by L α [φ](t) -Re iα -Re iα φ(ξ) e -tξ dξ ≤ |φ| Λ d α (t, )s ds + +∞ R e -d α (t, )s ds ≤ 2|φ| Λ e iα R d α (t, ) e -dα(t, )R ,whered α (t, ) = (e iα t) -Λ > 0, d α (t, ) = -( e iα πi √ ) -(e iα t) -Λ > 0,(63)andd α (t, ) = min{d α (t, ), d α (t, )}is the distance of t from the border of the strip T + α (Λ,√). Therefore to estimate the difference between ȳ+ (t,√ ) = L ᾱ[ῡ + ](t) and ỹ-(t, √ ) = L α[υ -](t) on T +ᾱ need only to estimate the difference between the truncated integrals.

1 e 2 e

 12 iα j ], ᾱ = α 0 < α 1 < . . . < α N -1 < α N = α, covering the double-sector α∈[ ᾱ, α] [-Re iα , Re iα ] that satisfy i) |φ j (ξ)| ≤ K Λ|ξ| 1+|e ξπi √ | , ξ ∈ U j , ii) |φ j-1 (ξ) -φ j (ξ)| ≤ K Λ|ξ|+(2Λ-λα)(R-|ξ|) 1+|e ξπi √ | for ξ ∈ [-Re iα , Re iα ] ⊂ U j-1 ∩ U j ,where λ α := -( e iα πi √ ).

e 0 e

 0 Λ|ξ|+(2Λ-λα)(R-|ξ|) 1 + |e ξπi √ | |e -tξ | d|ξ| ≤ R Λs+(2Λ-λα)(R-s)-(e iα t)s ds + -R 0 e -Λs+λαs+(2Λ-λα)(R+s)-(e iα t)s ds = 1 d α (e -d α R -e -(d α +d α )R ) + 1 d α (e -d α R -e -(d α +d α )R ) ≤ 2 dα e -dαR ,andRe i α Re i ᾱ e Λ|ξ| 1 + |e ξπi √ | |e -tξ | d|ξ| + -Re i α -Re i ᾱ e Λ|ξ| 1 + |e ξπi √ | |e -tξ | d|ξ| ≤ 2R(α -ᾱ) e -dR , where d = min α∈[ ᾱ, α] d α . Combining these estimates results in the estimate (19) for ỹ-(t, √ ) -ȳ+ (t, √ ) . The estimate ỹ+ (t, √ ) -ȳ-(t, √) is symmetric. The approximate solutions φ j are constructed as in the proof of Proposition 30 as a fixed point of the operator G + α j (44) but this time with the convolution * in the direction α = α j replaced by its symmetric truncation[φ * ψ] iα φ(ξ -s)ψ(s) ds, if ξ ∈ [0, Re iα ],Re iα +ξ -Re iα φ(ξ -s)ψ(s) ds, if ξ ∈ [-Re iα , 0],

0 e

 0 i) For φ, ψ : e iα R → C integrable and with bounded | • | Λ e iα R -norm|[φ * ψ] α (ξ) -[φ * ψ] R α (ξ)| ≤ 4 λα-2Λ |φ| Λ e iα R |ψ| Λ e iα R • e Λ|ξ|+(2Λ-λα)(R-|ξ|) 1 + |e ξπi √ | ii) If φ, ψ : [-Re iα , Re iα ] → C be integrable and bounded, with |φ(ξ)| ≤ K 1 e Λ|ξ| 1 + |e ξπi √ | , and |ψ(ξ)| ≤ K 2 e Λ|ξ|+(2Λ-λα)(R-|ξ|) 1 + |e ξπi √ | , then |[φ * ψ] R α (ξ)| ≤ 4K 1 K 2 R e Λ|ξ|+(2Λ-λα)(R-|ξ|) 1 + |e ξπi √ | Proof. i) If ξ ∈ [0, Re iα ] we can estimate +∞e iαRe iα e Λ(|ξ-s|+|s|)(1 + e (ξ-s)πi √ )(1 + e sπi √ ) ds ≤ +∞ R e Λ(2s-|ξ|)+λα(|ξ|-s) ds ≤ 2 λ α -2Λ • e Λ|ξ|+(2Λ-λα)(R-|ξ|) 1 + |e ξπi √ | ,and by symmetry s → ξ -s the same holds for the integral from -∞e iα toξ -Re iα .Similarlyfor ξ ∈ [-Re iα , 0]. ii) If ξ ∈ [0, Re iα ] we can estimate Re iα ξ-Re iα e Λ(|ξ-s|+|s|)+(2Λ-λα)(R-|s|) 2s-|ξ|)+λα(|ξ|-s)+(2Λ-λα)(R-s) ds + |ξ| Λ|ξ|+(2Λ-λα)(R-s) ds + 0 -R+|ξ| e -Λ(2s-|ξ|)+λαs+(2Λ-λα)(R+s) ds ≤ [(R -|ξ|) + |ξ| + (R -|ξ|)] e Λ|ξ|+(2Λ-λα)(R-|ξ|) ≤ 4R e Λ|ξ|+(2Λ-λα)(R-|ξ|) 1 + |e ξπi √ | , using that (2Λ -λ α ) < 0. Similarly for ξ ∈ [-Re iα , 0].Corollary 44. If υ ± i , i = 1, 2, are two solution of the convolution equations (43) of Proposition 36 on domains Ω i , then their difference is exponentially flat in √ : Λ|ξ|+(2Λ-λα)(R-|ξ|)

  +∞, and one can see that | B α [f ] Λ Ω 1 can be made arbitrarily small taking Λ sufficiently large (cf. Lemma 21).

	Lemma 42. Let φ χ + , ψ χ + be analytic functions on Ω 1 such that φ χ + (0,

An isolated singular point of a holomorphic vector field in C m+1 is of saddle-node type if its linearization matrix has exactly one zero eigenvalue, and it is of codimension 1 if the multiplicity of the singular point is 2.

If instead f (x, , 0) was only O(|x| + | |), and u ± √ ∈ C m were the unique solutions of 0 =

These α will later correspond to the direction of the unfolded Laplace integrals (8), andT ± α (Λ,√) to their strips of convergence.

More precisely to a covering space of the x-plane ramified at { √ , -√ }, the Riemann surface of t(x, ) (9).

√dξ = e -t(x,ξ 2 )•ξ dξ, in our notation.

Acknowledgements

I am very grateful to Christiane Rousseau for many helpful discussions and to Reinhard Schäfke and Loïc Teyssier for their interest in my work. The paper was prepared during my doctoral studies at Université de Montreal and finalized during my stay at Université de Strasbourg -I want to thank both institutions for their hospitality.