Ricardo Pérez-Marco
email: ricardo.perez-marco@imj-prg.fr

Keywords: 2000 Mathematics Subject Classification. 08A02, 30D99, 30F99 Eñe product, Hadamard product, singularities with monodromy, transalgebraic class

MONODROMIES OF SINGULARITIES OF THE HADAMARD AND E ÑE PRODUCT

RICARDO P ÉREZ-MARCO Abstract. We prove that singularities with monodromies are preserved by Hadamard product, and we find an explicit formula for the monodromy of the singularities of the Hadamard product. We find similar formulas for the eñe product whose monodromy is better behaved. With these formulas we give new direct proofs of classical results and prove the invariance of interesting rings of functions by Hadamard multiplication and eñe product. 1.1. Hadamard and eñe product. Given two power series

F (z) = A 0 + A 1 z + A 2 z 2 + . . . = n≥0 A n z n G(z) = B 0 + B 1 z + B 2 z 2 + . . . = n≥0 B n z n
their classical Hadamard product is the power series

F ⊙ G(z) = A 0 B 0 + A 1 B 1 z + A 2 B 2 z 2 + . . . = n≥0 A n B n z n
and their exponential eñe product is defined by

F ⋆ e G(z) = -A 1 B 1 z -2A 2 B 2 z 2 + . . . = - n≥0 nA n B n z n
The Hadamard product and the exponential eñe product are commutative internal operations on the additive group of formal power series C[[z]] (or A [[z]] for a commutative ring A), and (C[[z]], +, ⊙) and (C[[z]], +, ⋆ e ) are commutative rings. These products are also internal operations on the additive subgroup C{{z}} of power series with a positive radius of convergence, and (C{{z}}, +, ⊙) and (C{{z}}, +, ⋆ e ) are commutative subrings.

Hadamard (1899, [START_REF] Hadamard | Théorème sur les séries entières[END_REF]) proved the Hadamard Multiplication Theorem that locates the singularities of the principal branch of F ⊙ G which are products of singularities of F and G. The exponential eñe product has also a beautiful interpretation in terms of divisors, i.e. zeros and poles (2019, [START_REF] Érez-Marco | The eñe product for a commutative ring[END_REF]). More precisely, if a 0 = b 0 = 0 and we consider the exponential of the power series f (z) = exp(F (z)) = 1 + a 1 z + a 2 z 2 + . . . Thus the exponential eñe product is the bilinearization of the eñe product through the exponential. The universality of the expression of the coefficients (c n ) allows to define in general the eñe product over a commutative ring of coefficients A, more precisely, if A = 1 + A[[z]], then A is a group for the multiplication and when we adjoint the eñe product we obtain a commutative ring (A, ., ⋆).

If f and g are non-constant polynomials (note in particular that in thata case F and G are not polynomials), with respective roots (α) and (β) (counted with multiplicity), so that

f (z) = α 1 - z α g(z) = β 1 - z β
then we have the remarkable formula

f ⋆ g(z) = α,β 1 - z αβ .
This interpretation with zeros can be taken as the starting point of the theory of the eñe product as is done in [START_REF] Érez-Marco | The eñe product for a commutative ring[END_REF] (2019). Then the divisor property can be extended to entire and meromorphic functions on the plane, and even to transcendental singularities (2019, [START_REF] Érez-Marco | Eñe product in the transalgebraic class[END_REF]). The eñe product is both natural from the algebraic point of view, and for the analytic one (for example it is compatible with Hadamard-Weierstrass factorizations, see section 9 of [START_REF] Érez-Marco | Eñe product in the transalgebraic class[END_REF]). This property of zeros is the closely related Hadamard Theorem, although the Hadamard product has no such direct interpretation in terms of zeros. Indeed, we have a formula relating the exponential eñe product and the Hadamard product, namely

F ⋆ e G = -K 0 ⊙ F ⊙ G
where K 0 is the Koebe function

K 0 (z) = - n≥1 n z n = - z (1 -z) 2
For the definition of the eñe product and more algebraic and analytic properties and formulas we refer to [START_REF] Érez-Marco | The eñe product for a commutative ring[END_REF] where we extend the divisor interpretation to meromorphic functions with zeros and poles. In [START_REF] Érez-Marco | Eñe product in the transalgebraic class[END_REF] we extend further the eñe product to the transalgebraic class. For the Riemann sphere P 1 C, this transalgebraic class is composed by functions with a finite number of exponential singularities. These are of the form R 0 exp R 1 where R 0 and R 1 are rational functions. We prove in [START_REF] Érez-Marco | Eñe product in the transalgebraic class[END_REF] that the divisor interpretation still holds for exponential singularities, which is natural when we view these singularities à la Euler as zeros or poles of "infinite order". From this point of view, we naturally introduce "eñe poles" and then functions with singularities with non-trivial monodromies arise naturally, in particular the hierarchy of polylogarithms.

Therefore, it becomes natural to investigate the extension of the divisor interpretation of the eñe product to singularities with non-trivial monodromy, i.e. non-uniform singularities in the XIXth century terminology. A uniform transcendental singularity is an isolated singularity without monodromy.

Almost simultaneously to the discovery by Hadamard of his Theorem determining the location of the singularities of F ⊙ G, Émile Borel (1898, [START_REF] Borel | Sur les singularités des séries de Taylor[END_REF]) proved that if F and G have uniform isolated singularities, i.e. isolated without monodromy, then the singularities of F ⊙ G are isolated and uniform. This result is related to the action of the eñe product on exponential singularities. Borel also makes the vague, but on point, observation that the nature of the singularities of F ⊙ G only depends on the nature of the singularities of F and G. "...la nature du point singulier αβ ne dépend que de la nature des points singuliers α et β..." 1 ( É. [START_REF] Borel | Sur les singularités des séries de Taylor[END_REF] The goal of this article is to make very precise this statement, by giving, in some natural situations, what seems to be a new explicit formula for the monodromy of the singularities of F ⊙ G in terms of the monodromies of singularities of F and G.

1 "... the nature of the singular point αβ depends only on the nature of the singular points α and β..." 1.2. Holomorphic monodromy formulas. We need first some definitions and properties on monodromies.

Definition 1.1 (Monodromy of an isolated singularity). Let F be an holomorphic function with an isolated singularity α ∈ C. We denote F + the analytic continuation of F when winding around α once in the positive orientation (in a neighborhood without any other singularity). The monodromy of F at the point α ∈ C is

∆ α F = F + (z) -F (z) .
Example. The simplest and basic example of monodromy is given by the logarithmic function, that we normalize properly, at the isolated singularity α = 1,

∆ 1 1 2πi log z = 1
We consider in this article only holomorphic monodromies:

Definition 1.2 (Holomorphic monodromy). A function F with an isolated singularity at α ∈ C has a holomorphic monodromy at α when ∆ α F is holomorphic in a neighborhood of α.

Observe that if the monodromy is holomorphic then

∆ 2 α F = 0. When ∆ 2 α F = 0, if F 0 = F - 1 2πi log(z -α)∆ α F
then ∆ α F 0 = 0, and F 0 has a uniform singularity at α (isolated without monodromy). We have proved:

Proposition 1.3. When ∆ 2 α F = 0, we can write uniquely F as

F = F 0 + 1 2πi log(z -α)∆ α F
where F 0 has a uniform singularity at 0.

The singularity α is said to be totally holomorphic when ∆ α F and the germ F 0 are holomorphic at α.

(note the slight abusive notation since F 0 depends on the singularity α) Our main result computes the monodromy ∆ αβ (F ⊙ G) from the monodromies ∆ α F and ∆ β G when these are holomorphic singularities. We have a remarkable explicit formula: Theorem 1.4 (Holomorphic monodromy formula for the Hadamard product). We consider F and G holomorphic germs at 0 with respective set of singularities (α) and (β) in C. We assume that the singularities are isolated and holomorphic, that is, ∆ α F , resp. ∆ β G, is holomorphic at α, resp. β. Then the set of singularities of the principal branch of F ⊙ G is contained in the product set (γ) = (αβ) and is composed by isolated singularities which are holomorphic, and we have the formula

∆ γ (F ⊙ G)(z) = - α,β αβ=γ Res u=α F 0 (u)∆ β G(z/u) u - α,β αβ=γ Res u=β G 0 (u)∆ α F (z/u) u (1) - 1 2πi α,β αβ=γ z/β α ∆ α F (u)∆ β G(z/u) du u
When the singularities are totally holomorphic, we have the simpler formula

(2) ∆ γ (F ⊙ G)(z) = - 1 2πi α,β αβ=γ z/β α ∆ α F (u)∆ β G(z/u) du u
Notice the exceptional situation at z = 0: we can have that the monodromies of F and G are holomorphic at z = 0 but the monodromy of the Hadamard product can have a singularity at z = 0 with a non-trivial monodromy that is generated by the term 1/u in the integrand of the monodromy convolution formula (1). This happens for example for polylogarithms (see section 6.1).

We observe that when ∆ α F = ∆ β G = 0 for all singularities α and β, then the singularities are all holomorphic, ∆ γ (F ⊙ G) = 0, and Borel's Theorem is a direct Corollary of our formula.

Borel also observes that when F is a rational function, then one can construct a differential operator D F such that the the singularities of αβ F ⊙ G are "of the same nature" as those of D F G at β. The holomorphic monodromy formula makes renders this explicit when G has holomorphic singularities.

Corollary 1.5. Let F be a meromorphic function in C (for example a rational function), holomorphic at 0, with set of poles (α), and G with totally holomorphic singularities (β). The monodromies of F ⊙ G are in the differential ring generated by the ∆ b G(z/α), with field of constants generated by the coefficients of the polar parts of F . More precisely, consider the polar part of F at each pole α

d k=1 a k,α (u -α) k , then we have ∆ γ (F ⊙ G) = - α,β,k αβ=γ a k,α (k -1)! d k-1 du k ∆ β G(z/u) u u=α Example.
A simple example occurs when we take F = -K 0 , where K 0 (z) = z/(1 -z) 2 is the Koébé function wich has a simple pole of order 2 at z = 1 and polar part

-K 0 (z) = - 1 z -1 - 1 (z -1) 2 .
Then we compute

∆ β (-K 0 ⊙ G) = -Res u=1 - 1 u -1 - 1 (u -1) 2 ∆ β G(z/u) u = [∆ β G(z/u)/u] u=1 + ∆ β G ′ (z/u) u - ∆ β G(z/u) u 2 u=1 = ∆ β G(z) + ∆ β G ′ (z) -∆ β G(z) = ∆ β G ′ (z)
The conclusion is that the monodromies of the singularities of the Hadamard product with the negative of the Koebe function are the derivatives of the monodromies.

We have a similar formula for the exponential eñe product.

Theorem 1.6 (Holomorphic monodromy formula for the exponential eñe product). We consider F and G holomorphic germs at 0 with respective sets of singularities (α) and (β). We assume that the singularities are isolated and holomorphic. Then the set of singularities of the principal branch of F ⋆ e G is contained in the product set (γ) = (αβ) and is composed by isolated singularities with holomorphic monodromies, and we have

∆ γ (F ⋆ e G)(z) = α,β αβ=γ Res u=α (F ′ 0 (u)∆ β G(z/u)) + α,β αβ=γ Res u=β (G 0 (u)∆ α F ′ (z/u)) (3) + 1 2πi α,β αβ=γ ∆ α F (α)∆ β G(z/α) + 1 2πi α,β αβ=γ z/β α ∆ α F ′ (u)∆ β G(z/u) du
When the singularities are totally holomorphic, we have the simpler formula

(4) ∆ γ (F ⋆ e G)(z) = 1 2πi α,β αβ=γ ∆ α F (α)∆ β G(z/α) + 1 2πi α,β αβ=γ z/β α ∆ α F ′ (u)∆ β G(z/u) du
Notice that the absence of the factor 1/u in the integral of the monodromy convolution formula (3) for the exponential eñe product generates no extra singularities at z = 0 for non-principal branches of F ⋆ e G. This explains why the monodromies of singularities of the exponential eñe product have a better analytic behavior than those for the Hadamard product. The symmetry of the formula on F and G is clear in the first line, and for the second line it follows by integration by parts using basic properties of the monodromy operator ∆ α (see Proposition 5.3).

For the exponential eñe product we have the same Borel's type of Theorem. It follows from formula [START_REF] Biswas | Caratheodory convergence of log-Riemann surfaces and Euler's formula, Geometry, groups and dynamics[END_REF] We have also an analogue of Corollary 1.5. The result is stronger because of the better analytic properties of the eñe product.

that if ∆ α F = ∆ β G = 0 then ∆ γ (F ⋆ e G) =
Corollary 1.8. Let F be a function in C with a discrete set (α) of singularities with constant monodromies, such that the germs F 0 are meromorphic, F is holomorphic at 0, and G with totally holomorphic singularities (β). The monodromies of F ⋆ e G are in the differential ring generated by the ∆ b G(z/α), with field of constants generated by the coefficients of the polar parts of F and the constants ∆αF 2πi . More precisely, consider the polar part of each F 0 at each pole α

d k=1 a k,α (u -α) k , then we have ∆ γ (F ⊙ G) = - α,β,k αβ=γ a k,α (k -1)! d k du k (∆ β G(z/u)) u=α + α,β αβ=γ ∆ α F (α) 2πi ∆ β G(z/α)
A Corollary of the algebraic nature of these monodromy formulas is the invariance by the Hadamard and eñe product of some natural rings of functions. Consider a field K ⊂ C with 2πi ∈ K, and consider the ring P M L(K) (polynomial logarithmic monodromy ring with coefficients in K) of germs holomorphic at 0 with only isolated singularities with monodromy in K[z, log z].

Corollary 1.9. The P LM (K) ring is closed under Hadamard, resp. eñe, product, and is the minimal Hadamard, resp. eñe, ring containing the class of functions with polynomial monodromies in K[z] with isolated singularities at points of K.

Convolution formulas.

The main tool in the proof of Hadamard Theorem is Pincherle (or Hadamard) convolution formula (1885, see [START_REF] Pincherle | Note sur une intégrale définie[END_REF], [START_REF] Laurent | Traité d'Analyse[END_REF] and [START_REF] Hadamard | Théorème sur les séries entières[END_REF]):

Proposition 2.1 (Pincherle convolution formula). The Hadamard product has the integral form

(5) F ⊙ G(z) = 1 2πi η F (u)G(z/u) du u
where η is a positively oriented circle centered at 0 of radius r > 0 with |z|/R G < r < R F , where R F and R G are the respective radii of convergence of F and G, so that F (u) and G(z/u) are well defined.

We can take for η any Jordan curve located in the annulus bounded by the circles of radii R F and |z|/R G and with winding number +1 with respect to 0.

Proof. The convolution formula immediately follows from the integration term by term of the series

F (u)G(z/u) u = n,m≥0 F n G m u n-m-1 z m
and the application of Cauchy formula

1 2πi η u n-m-1 du = δ n,m
where δ n,m denotes the Kronecker symbol.

For the exponential eñe product we have a similar convolution formula:

Proposition 2.2 (Exponential eñe product convolution formula). The exponential eñe product has the integral form

(6) F ⋆ e G(z) = - 1 2πi η F ′ (u)G(z/u) du
where η is a circle centered at 0 with the same conditions as before.

Proof. The proof is similar observing that R F ′ = R F and integrating term by term

F ′ (u)G(z/u) = n,m≥0 nF n G m u n-m-1 z m
and using Cauchy formula as before.

The convolution formula [START_REF] Biswas | Uniformization of higher genus finite type log-Riemann surfaces[END_REF] gives the analytic continuation of the Hadamard product F ⊙ G using the analytic continuation of F and G. We only need to deform homotopically the contour η when we move z around. The only obstruction to this continuation follows from a close inspection of the convolution formula: F ⊙ G(z) does extend analytically unless z = 0 (except for the principal branch) or when we hit a point z ∈ C such that we have u ∈ C with both conditions u = α z/u = β with α and β singularities of F and G respectively. This happens if and only if z = αβ. Thus we have proved Hadamard Theorem: Theorem 2.3 (Hadamard, 1899, [START_REF] Hadamard | Théorème sur les séries entières[END_REF]). The singularities of the principal branch of F ⊙ G are of the form γ = αβ where α and β are singularities of F and G respectively.

The origin 0 is not a singularity of the principal branch by assumption. But the convolution formula shows that the origin 0 can become a singularity of other branches of the analytic continuation of F ⊙ G because of the 1/u factor in the integrand. Moreover, the convolution formula also proves that if F and G are fluent in the sense of Liouville and Ritt, which, roughly speaking, means that the functions have an analytic extension around singularities, then their Hadamard product F ⊙ G is fluent. The definition of fluency given by Ritt in his book 2 (1948, [START_REF] Ritt | Integration in finite terms. Liouville's theory of elementary methods[END_REF]) on Liouville theory of integration on finite terms (1833, [START_REF] Liouville | Mémoire sur les transcendantes elliptiques considérées comme fonction de leur amplitude[END_REF], [START_REF] Liouville | Mémoire sur l'intégration d'une class de fonctions transcendantes[END_REF]) is not precise and indeed there exists more or less stronger versions of fluency (see section 5.4.1 of [START_REF] Khovanski | Topological Galois Theory[END_REF], and in particular [START_REF] Levagnini | [END_REF]). Fluency is a key property of functions in the old Liouville classification of transcendental functions.

With these considerations, we can state the following geometric improvement of Hadamard Theorem (the reader, if not familiar, can skip this Theorem where we use the language of log-Riemann surfaces, see [START_REF] Biswas | Tube-log Riemann surfaces[END_REF] for general background, [START_REF] Biswas | Caratheodory convergence of log-Riemann surfaces and Euler's formula, Geometry, groups and dynamics[END_REF] for more general definitions, and [START_REF] Biswas | Uniformization of simply connected finite type log-Riemann surfaces, Geometry, groups and dynamics[END_REF] and [START_REF] Biswas | Uniformization of higher genus finite type log-Riemann surfaces[END_REF] for further properties).

Theorem 2.4. Let S F and S G be the log-Riemann surfaces of the germs at 0 defined by F and G respectively. Then the log-Riemann surface S F ⊙G has a ramification set R F ⊙G such that

π F ⊙G (R F ⊙G ) ⊂ {0} ∪ (π F (R F ).π G (R G ))
where π F ⊙G : S F ⊙G → C, π F : S F → C and π G : S G → C are the canonical projections, and A.B denotes the set of all products ab with a ∈ A and b ∈ B.

In particular, if the ramification sets R F and R G are discrete, then R F ⊙G is discrete.

For the exponential eñe product we have an analogue of Hadamard Theorem. We notice first that the singularities of F and F ′ are the same (local derivative and primitive preserve holomorphy). Hence we obtain: Theorem 2.5 (Singularities of the exponential eñe product). The singularities of the principal branch of F ⋆ e G are of the form γ = αβ where α and β are singularities of F and G respectively.

We have a corresponding Theorem for the log-Riemann surface S F ⋆eG which is simpler since we don't need to add the origin in the locus of the projection. Theorem 2.6. Let S F and S G be the log-Riemann surfaces of the germs at 0 defined by F and G respectively. Then the log-Riemann surface S F ⋆eG has a ramification set R F ⋆eG such that

π F ⋆eG (R F ⋆eG ) ⊂ π F (R F ).π G (R G )
where π F ⋆eG : S F ⋆eG → C, π F : S F → C and π G : S G → C are the canonical projections, and A.B denotes the set of all products ab with a ∈ A and b ∈ B.

In particular, if the ramification sets R F and R G are discrete, then R F ⋆eG is discrete. Ritt's book is from the pre-differential algebra era.

Monodromy of singularities and monodromy operator.

The inspection of the convolution formula shows also that if both F and G have isolated singularities with monodromy then the singularities of F ⊙ G and of F ⋆ e G are also isolated with monodromy (including the case of trivial monodromy). More precisely, we recall the definition of the monodromy and we define the monodromy operator: Definition 3.1 (Monodromy and operator monodromy of an isolated singularity). Let F be an holomorphic function with an isolated singularity α ∈ C. We denote F + the analytic continuation of F when turning around α once in the positive orientation (in a neighborhood without any other singularity). The monodromy of F at the point α ∈ C is

∆ α F = F + (z) -F (z) .
This definition can be extended also to regular points α of F . The map ∆ α defines a linear operator, the monodromy operator at α, on the vector space V α of holomorphic functions having a regular point or an isolated singularity at α. We also define the operator Σ α such that

Σ α F = F + and we have Σ α = I + ∆ α
In general, we define that monodromy operator along a path γ, σ γ : (C, γ(0)) → (C, γ(1)), which associates to a holomorphic germ at γ(0) its Weierstrass holomorphic continuation along γ at γ(1).

In a domain where the germ is holomorphic, σ γ = σ [γ] only depends on the homotopy class [γ] of γ, and for a loop γ with winding number 1 with respect to α, we have

Σ α = σ [γ] .
Note that non-locally, ∆ α F can develop singularities elsewhere at other points distinct from α. The structure of the monodromy at α is important, and justifies the following definitions. Definition 3.2. A function F with an isolated singularity at α ∈ C has a holomorphic, resp. meromorphic, uniform, monodromy at α when ∆ α F is holomorphic, resp. meromorphic, with a uniform isolated singularity at α, in a neighborhood of α. The singularity α is totally holomorphic if both ∆ α F and

F 0 = F -1 2πi log(z -α)∆ α F are holomorphic.
Recall (see Proposition 1.3) that when ∆ 2 α F = 0 we can write uniquely in a neighborhood of α

F = F 0 + 1 2πi log(z -α)∆ α F
where F 0 has a uniform isolated singularity at α.

The growth behavior of F near the singularity α is important.

Definition 3.3. Let F be function with a regular point or an isolated singularity at α ∈ C. Then F is integrable at α, or α is an integrable singularity, if we have

S-lim z→α (z -α) F (z) → 0
where this is a Stolz limit, that is, z → α with arg(z -α) bounded. The singularity α has order ρ > 0, if F (z) = O(|z -α| -ρ ) in any Stolz angle 3 .

3

These are also called "singularities with moderate growth in sectors", see [START_REF] Sauloy | Differential Galois theory through Riemann-Hilbert correspondence[END_REF], section 9.2, or "regular singularities" in the context of solutions of differential systems, see [START_REF] Anosov | The Riemann-Hilbert problem[END_REF], p.8.

Note that integrable, order ρ and finite order functions at α is a vector space. These vector spaces are O α -modules, where O α is the local ring of holomorphic germs at α. For an integrable singularity we have α+ǫ α F (u) du → 0 and η F (u) du → 0 when ǫ ∈ C is small and ǫ → 0, and the loop η is a small circle around α with radius → 0. Adapting the arguments given in [START_REF] Jungen | Sur les séries de Taylor nayant que des singularit és algébrico-logarithmiques sur leur cercle de convergence[END_REF] we can prove that the Hadamard product of integrable singularities is integrable, but we don't need this result in this article. Proposition 3.4. If F is holomorphic in a pointed neighborhood of α, i.e. ∆ α F = 0, and integrable, then F is holomorphic at α.

Proof. We can write a converging Laurent expansion in a pointed neighborhood of α,

F (z) = n∈Z a n (z -α) n
and we have the Cauchy formula for the coefficients

a n = 1 2πi η f (u) (z -u) n du .
When we shrink η to α, the integrability estimates show that a n = 0 for all n < 0, hence F is holomorphic.

Proposition 3.5. We have

∆ α (F.G) = ∆ α (F ).G + F.∆ α (G) + ∆ α (F ).∆ α (G)
Proof. It follows by analytic continuation that Σ α (F.G) = Σ α (F ).Σ α (G) and using Σ α = I + ∆ α gives the result. Corollary 3.6. If ∆ 2 α (F ) = 0, in particular when the monodromy of F at α is holomorphic or meromorphic, then

∆ α F - 1 2πi log(z -α) ∆ α (F ) = 0
In that case we can write,

F (z) = F 0 (z) + 1 2πi log(z -α) ∆ α (F )
where ∆ α F 0 = 0.

Proof. We observe that

∆ α 1 2πi log(z -α) = 1
and then from the previous Proposition we get

∆ α F - 1 2πi log(z -α) ∆ α (F ) = ∆ α (F ) -1.∆ α (F ) - 1 2πi log(z -α).∆ 2 α (F ) -1.∆ 2 α (F ) = ∆ α (F ) -∆ α (F ) -0 -0 = 0
Corollary 3.7. We assume that ∆ 2 α F = 0. The singularity α of F is integrable if and only if the singularity is totally holomorphic, that is ∆ α F and F 0 are holomorphic at α.

Proof. The condition is necessary. If F is integrable at α, from the definition we get that F + is integrable, hence ∆ α F = F + -F is integrable and holomorphic in a pointed neighborhood of α. Proposition 3.4 implies that ∆ α F is a holomorphic function at α. Then also

1 2πi log(z -α) ∆ α (F ) is integrable at α, hence F 0 (z) = F (z) -1 2πi log(z -α) ∆ α (F
) is also integrable. Again, by Proposition 3.4, F 0 being holomorphic in a pointed neighborhood and integrable, it is holomorphic.

The condition is sufficient. We assume that F 0 and ∆ α F are both holomorphic. We have that 1 2πi log(z -α) ∆ α (F ) is integrable because r log r → 0 when r → 0. Also, F 0 is integrable since is it holomorphic. Now, adding these two integrable functions we get that F is integrable. Now, observe that we have

F + = F + ∆ α F , hence (F + ) ′ = F ′ + (∆ α F ) ′
and since (F ′ )+ = (F + ) ′ , which is obvious by analytic continuation, then we conclude that the monodromy operator ∆ α commutes with the derivation. Proposition 3.8. We have

∆ α (F ′ ) = (∆ α F ) ′ Corollary 3.9. Let ϑ = P (D) = N n=0 a n D n with D = d/dz be a differential operator, with P ∈ C[X]. We have ∆ α (ϑF ) = ϑ(∆ α F
) . Therefore, if F satisfies the differential equation ϑF = 0 then the monodromy ∆ α F satisfies the same differential equation ϑ(∆ α F ) = 0 and ∆ α F belongs to the finite dimensional vector space of solutions. We have the same result for a system of differential equations.

The monodromy ∆ α F can have an isolated singularity at α with non-trivial monodromy, and for example this happens at singular points of differential equations, and of course for algebraic functions. For these we have the simple, but important following observation: Proposition 3.10. We assume that F is an algebraic function satisfying the algebraic equation

P (z, F ) = 0 with P ∈ C[x, y].
Then ∆ α F is an algebraic function satisfying an algebraic equation of lower degree

Q(z, ∆ α F ) = 0 with Q ∈ C[x, y] and deg y Q < deg y P Proof.
By analytic continuation around α we have

P (z, F + ∆ α F ) = 0
developing the powers of F + ∆ α F and using P (z, F ) = 0 we get the explicit expression for Q that has a lower degree in the second variable.

We will also need the following change of variables formula. Proposition 3.11 (Change of variables formula). Let ϕ : (C, β) → (C, α) a local holomorphic diffeomorphism, α = ϕ(β). We have

∆ ϕ -1 (α) (F • ϕ) = ∆ α (F ) • ϕ .
Proof. We consider a local loop γ enclosing α with winding number 1 with respect to α. Its preimage ϕ -1 (γ) has winding number 1 with respect to β. When we continue along ϕ -1 (γ) the return analytic continuation of F • ϕ is F • ϕ + ∆ β (F • ϕ) by definition of the monodromy. The return analytic continuation of F along γ is F + ∆ α F , hence we have

F • ϕ + ∆ β (F • ϕ) = (F + ∆ α (F )) • ϕ = F • ϕ + ∆ α (F ) • ϕ
and we get the stated formula.

Corollary 3.12. Let z 0 ∈ C * , then we have

∆ β (F (z 0 /z)) = ∆ z0/β (F )(z 0 /z) .
Proof. With the change of variables ϕ(z) = z 0 /z, thus ϕ -1 (z) = z 0 /z, we get the result using Proposition 3.11.

Note that the formula in the Corollary remains valid for the monodromy at ∞ in the Riemann sphere, α = ∞ ∈ C. Thus, the formula holds in general for a Moëbius transformation ϕ.

This result is a particular case for n = 1 (local diffeomorphism) of the local degree n ≥ 1 case: Proposition 3.13. Let ϕ : (C, β) → (C, α) a local holomorphic map with β a critical point of degree n ≥ 1 (hence ϕ is of local degree n). We have

∆ α (F ) • ϕ = n k=1 n k ∆ k β (F • ϕ)
Lemma 3.14. Let γ be a local loop with winding number n ≥ 1 with respect to β ∈ C in a neighborhood where β is the only singularity of F in this neighborhood. We denote ∆

(n)

β F the monodromy of F along γ that only depends on the homotopy class [γ], i.e. on its winding number n ≥ 1. We have

∆ (n) β F = n k=1 n k ∆ k β F .
Proof. We observe that ∆

(n) β = σ n n.[γ] -I = Σ n β -I . Since Σ β = I + ∆ β ,
the result follows from Newton binomial formula.

Proof of Proposition 3.13. We carry out the same proof as before, but this time the pre-image ϕ -1 (γ) has winding number n ≥ 1 with respect to β = ϕ -1 (α). When we continue along γ the return analytic continuation of F is F + ∆ α (F ) by definition of the monodromy. The return analytic continuation of

F • ϕ along ϕ -1 (γ) is F • ϕ + ∆ (n) β (F • ϕ) hence we have F • ϕ + ∆ α (F ) • ϕ = F • ϕ + ∆ (n) β (F • ϕ) so ∆ α (F ) • ϕ = ∆ (n)
β (F • ϕ) and the formula follows from Lemma 3.14.

We leave the general study of monodromies ∆ α F having an isolated singularity at α with nontrivial monodromy to future articles.

Totally holomorphic monodromy formula.

As a preparation for the general case, we first prove the monodromy formula 2 in the totally holomorphic or, equivalently, the integrable case. Thanks to the integrability conditions, there is no local contributions in the integration path argument that is central in the proof. In the next section we treat the general case.

4.1.

Proof of the totally holomorphic monodromy formula: Single singularity case. We prove in this section Theorems 1.4 and 1.6 in the totally holomorphic case. The proof of Theorem 1.6 is similar and indications will be given at the end, so we concentrate in the proof of 1.4.

The first observation is that the result is purely local near the singularities. We may have some multiplicity when there exists distinct pairs (α, β) = (α ′ , β ′ ) such that αβ = α ′ β ′ . Then we have a linear superposition of the different contributions. To be precise, there is no need to assume in the Theorem that all singularities have the same holomorphic structure but only those α's and β's that contribute to the location γ.

We consider first the case where there is no multiplicity, so γ = αβ for a single pair (α, β), and we can assume that α = β = γ = 1, and the radii of convergence are R F = R G = 1. The general case is a superposition of this case. We consider a current value z close to 1, and we follow the analytic extension of F ⊙ G when z, starting at z 0 , turns around 1 once in the positive direction. We can start at z 0 with |z 0 | < 1, and z 0 close to 1. We start integrating Hadamard convolution formula on a circle η z0 of radius r with |z 0 | < r < 1, so that both series giving F (u) and G(z 0 /u) are converging. When z, starting at z 0 , moves around the point 1, we deform homotopically the integration path into η z so that z never crosses the integration path. With such condition, the convolution formula yields the analytic continuation of F ⊙ G. When we return to z 0 , the path is deformed into ηz0 (see Figure 2) the Hadamard product takes the value

(F ⊙ G) + (z 0 ). The difference ∆ 1 (F ⊙ G)(z 0 ) = (F ⊙ G) + (z 0 ) -(F ⊙ G)(z 0 )
is the monodromy at 1. According to Hadamard convolution formula, this difference can be computed by integrating on the homotopical difference of the two paths ηz0 -η z0 (Figure 2). Considering the intersection point a = [z 0 , 1] ∩ η z0 , and shrinking the path as shown in Figure 2, this difference is composed by four (indeed only two, repeated twice) vertical segments (it is indeed a "train track")

η 1 = [a, 1], η 2 = [a, z 0 ], η 3 = [a, 1
] and η 4 = [a, z 0 ] where we integrate in both directions different functions. Note also that the small turning loops around 1 and z 0 give no contribution when they shrink because of the integrability condition (in the proof of the general case there is a non-trivial residual contribution). We decompose further each path η j into two consecutive paths η j = η - j ∪ η + j so that the difference ηz0 -η z0 decomposes as

ηz0 -η z0 = η - 1 ∪ η + 1 ∪ η - 2 ∪ η + 2 ∪ η - 3 ∪ η + 3 ∪ η - 4 ∪ η + 4
where the paths η ± j follow each other in the order this union is written. We first list the functions that are integrated against the differential du/u in each path. To compute these functions, notice that the monodromy around u = 1 of F (u) is ∆ 1 F (u), and, according to Corollary 3.12, the monodromy around

z 0 of G(z 0 /u) is ∆ 1 G(z 0 /u), i.e. ∆ z0 (G(z 0 /u)) = (∆ 1 G)(z 0 /u) .
We need to take into account the sign corresponding to the orientation of the loop around each singularity. Note that we are assuming that the monodromies ∆ 1 F and ∆ 1 G are holomorphic and, in particular, there is no second monodromy at 1.

η - 1 →F (u)G(z 0 /u) η + 1 →F (u)G(z 0 /u) + ∆ 1 F (u)G(z 0 /u) η - 2 →F (u)G(z 0 /u) + ∆ 1 F (u)G(z 0 /u) η + 2 →F (u)G(z 0 /u) + F (u)∆ 1 G(z 0 /u) + ∆ 1 F (u)G(z 0 /u) + ∆ 1 F (u)∆ 1 G(z 0 /u) η - 3 →F (u)G(z 0 /u) + F (u)∆ 1 G(z 0 /u) + ∆ 1 F (u)G(z 0 /u) + ∆ 1 F (u)∆ 1 G(z 0 /u) η + 3 →F (u)G(z 0 /u) -∆ 1 F (u)G(z 0 /u) + F (u)∆ 1 G(z 0 /u) -∆ 1 F (u)∆ 1 G(z 0 /u)+ + ∆ 1 F (u)G(z 0 /u) + ∆ 1 F (u)∆ 1 G(z 0 /u) = F (u)G(z 0 /u) + F (u)∆ 1 G(z 0 /u) η - 4 →F (u)G(z 0 /u) + F (u)∆ 1 G(z 0 /u) η + 4 →F (u)G(z 0 /u) -F (u)∆ 1 G(z 0 /u) + F (u)∆ 1 G(z 0 /u) = F (u)G(z 0 /u)
Now, for the contributions of each integral we have

ηj = η - j - η + j
and these contributions for each integral η j are, respectively,

η1 -∆ 1 F (u)G(z 0 /u) du u = - 1 a ∆ 1 F (u)G(z 0 /u) du u η2 (-F (u)∆ 1 G(z 0 /u) -∆ 1 F (u)∆ 1 G(z 0 /u)) du u = - z0 a (F (u)∆ 1 G(z 0 /u) + ∆ 1 F (u)∆ 1 G(z 0 /u)) du u η3 (∆ 1 F (u)G(z 0 /u) + ∆ 1 F (u)∆ 1 G(z 0 /u)) du u = 1 a (∆ 1 F (u)G(z 0 /u) + ∆ 1 F (u)∆ 1 G(z 0 /u)) du u η4 F (u)∆ 1 G(z 0 /u) du u = 1 a F (u)∆ 1 G(z 0 /u) du u
Adding up, after some cancellations when pairing η 1 and η 3 , and pairing of η 2 and η 4 , we get

η1 + η3 = 1 a ∆ 1 F (u)∆ 1 G(z 0 /u) du u η2 + η4 = - z0 a ∆ 1 F (u)∆ 1 G(z 0 /u) du u
and finally

ηz 0 -ηz 0 = η1 + η2 + η3 + η4 = - z0 1 ∆ 1 F (u)∆ 1 G(z 0 /u) du u
which gives the formula for the case of a single singularity.

Note that in this path deformation argument, there is no residual contribution contribution at 1 because the functions and their monodromies are all integrable at 1. 4.2. Proof for higher multiplicity singularities. The initial Jordan loop of integration η z0 separates the singularities (α) of F , which are in the outside unbounded region, from the singularities (z 0 /β) of G(z 0 /u), which are in the inside bounded region. As before, when the point z starting at z 0 circles once around αβ = γ, then all points z/β circles once around the corresponding α in a synchronized choreography. We end-up with a path ηz0 as shown in Figure 1 (reversing the orientation). Then the difference of paths η z0 -ηz0 is decomposed into a finite number of quadruple loops as the one considered before, one for each pair (α, β) such that γ = αβ. The total contribution adds the contribution of each quadruple loop and the formula follows.

5. General holomorphic monodromy formula.

5.1.

Formula for the Hadamard product. We prove the general result by following the ideas from the previous section. We are reduced totreat the case of a singularity αβ without multiplicity, and we can assume α = β = 1. We consider the same integration path.

The only difference appears when we shrink loops at u = 1 and u = z 0 . For example, for the first path η 1 , when we turn around u = 1, we get an extra contribution of lim

ǫ1→1 1 2πi ǫ1 F (u)G(z 0 /u) du u
In the case of a general singularity γ = αβ, this residue contribution is

R = -Res u=α F 0 (u)∆ β G(z 0 /u) u -Res u=β G 0 (u)∆ α F (z 0 /u) u
and this gives the holomorphic monodromy formula (1).

5.2. Formula for the exponential eñe product. The proof for the exponential eñe product follows the same lines. The only difference is in the the sign in the convolution formula and the integrating function. Since the monodromy operator commutes with differentiation there is almost no difference in the entire argument. We need to use the following Lemma:

Lemma 5.2. At a holomorphic singulartiy α we have

(F ′ ) 0 = F ′ 0 + 1 2πi 1 z -α ∆ α F
Proof. We use the commutation of the monodromy operator with the differentiable operator. We differentiate the decomposition

F (z) = F 0 (z) + 1 2πi log(z -α)∆ α F (z)
and obtain

F ′ (z) = F ′ 0 (z) + 1 2πi 1 z -α ∆ α F (z) + 1 2πi log(z -α)∆ α F ′ (z)
and by uniqueness of the monodromy decomposition for F ′ from Proposition 1.3, we get the result.

The same proof as before, starting from the exponential eñe product convolution formula, gives

∆ γ (F ⋆ e G)(z) = α,β αβ=γ Res u=α ((F ′ ) 0 (u)∆ β G(z/u)) + α,β αβ=γ Res u=β (G 0 (u)∆ α F ′ (z/u)) + 1 2πi α,β αβ=γ z/β α ∆ α F ′ (u)∆ β G(z/u) du
Now, using the Lemma 5.2, we have

Res u=α ((F ′ ) 0 (u)∆ β G(z/u)) = Res u=α (F ′ 0 (u)∆ β G(z/u)) + 1 2πi Res u=α 1 u -α ∆ α F (u)∆ β G(z/u) = Res u=α (F ′ 0 (u)∆ β G(z/u)) + 1 2πi ∆ α F (α)∆ β G(z/α)
and formula (3) follows.

This commutation property for the derivation is also the reason for the symmetry on F and G of the formula. By integration by parts we have: Proposition 5.3. For γ = αβ we have

∆ α F (α)∆ β G(z/α) + z/β α ∆ α F ′ (u)∆ β G(z/u) du = ∆ α F (z/β)∆ β G(β) + z γ ∆ β G ′ (u)∆ α F (z/u) du
Proof. We perform an integration by parts, and the change of variables v = z/u,

z/β α ∆ α F ′ (u)∆ β G(z/u) du = [∆ α F (u)∆ β G(z/u)] z/β α - z/β α ∆ α F (u)∆ β G ′ (z/u) - 1 u 2 du = [∆ α F (u)∆ β G(z/u)] z/β α - z/α β ∆ α F (z/v)∆ β G ′ (v) dv
We could have derived also the formula for the monodromy of the exponential eñe product from the formula from the Hadamard product by using

F ⋆ e G = -K 0 ⊙ F ⊙ G .
It is instructive to derive it. We have already seen that the effect on the monodromy of the Hadamard product with -K 0 is just the derivation, hence

∆ γ (F ⋆ e G) = (∆ γ F ⊙ G) ′
and we only need to derivate formula 1. Using the commutation with the monodromy operator and previous Lemma 5.2 we get the result again.

The proof of Corollary 1.8 is straightforward from the eñe monodromy formula. 6. Applications.

6.1. Application 1: Monodromy of polylogarithms. We use our formula to compute the classical monodromy of polylogarithms, that is done in the literature using functional equations (as for example in [16]). The polylogarithm Li k (z) for k = 1, 2, . . . can defined for |z| < 1 by the converging series

Li k (z) = +∞ n=1 n -k z n
Therefore Li 1 is given by the logarithm, Li 1 (z) = -log(1 -z) that has principal branch with a multivalued holomorphic extension to C -{1} with a unique singularity at 1 which is of logarithmic type with a constant monodromy ∆ 1 Li 1 = -2πi .

Higher polylogarithms can also be defined inductively by integration

Li k+1 (z) = z 0 Li k (u) du u
as we readily see integrating term by term the defining power series in its disk of convergence. Using the integral expression we prove by induction that their principal branch extends holomorphically to a multivalued function on C -{1} with a unique singularity at 1 (non-principal branches can have also singularities at 0 as we will see).

From the power series definition it is clear that higher order polylogarithms can also be defined inductivelly using Hadamard multiplication

Li k+1 = Li k ⊙ Li 1
The minimal Hadamard ring structure (for the sum and the Hadamard product) generated by the logarithm is obtained by adjoining higher order polylogarithms, for k, l ≥ 1,

Li k+l = Li k ⊙ Li l
As a Corollary of our monodromy formula we can compute directly the monodromy at s = 1. Corollary 6.1. For k ≥ 2 the only singularities of the analytic continuation of Li k are located at 0 and 1. For k ≥ 1 the monodromy at 1 of the principal branch of Li k is holomorphic at z = 1 and, more precisely,

∆ 1 Li k = - 2πi (k -1)! (log z) k-1
Proof. For k = 1 it is the monodromy of the classical logarithm that is holomorphic (constant) at 1. Assuming by induction the result for k ≥ 1, the monodromy for Li k is holomorphic at z = 1, and we can use Theorem 1.6 with the formula Li k+1 = Li k ⊙ Li 1 , obsrving that the singularity at α = 1 is totally holomorphic, so we use formula (2), and we get (using the change of variables v = log u)

∆ 1 Li k+1 (z) = - 1 2πi z 1 - 2πi (k -1)! (log u) k-1 (-2πi) du u = - 2πi (k -1)! log z 0 v k-1 dv = - 2πi k! (log z) k
We can use the formula for the exponential eñe product to cross check this result since it follows from the power series expansion that, for k, l ≥ 1, Li k ⋆ e Li l = -Li k+l-1 and in particular,

Li k+1 = -Li k ⋆ e Li 2 Li 1 = -Li 1 ⋆ e Li 1
hence the minimal exponential eñe ring containing Li 1 is just generated by Li 1 , but the one containing Li 1 and Li 2 is generated by all the higher polylogarithms as before. Related to this we observe that we need to know the monodromy of Li 2 to start the induction with the exponential eñe product formula. The exponential eñe convolution formula and the Hadamard convolution formula are directly related using Li

′ k+1 (z) = Li k (z)/z since Li k ⋆ e Li l = 1 2πi η Li ′ k (u) Li l (z/u) du = 1 2πi η Li k (u) Li l (z/u) du u = -Li k ⊙ Li l = -Li k+l
Now, using the formula (4) for the eñe monodromy for the totally holomorphic singularity at α = 1, and the change of variables v = log u/ log z, we have

∆ 1 (Li k ⋆ e Li l ) = 1 2πi ∆ 1 Li k (1) Li l (z) + 1 2πi z 1 -2πi (k -1)! (log u) k-1 ′ -2πi (l -1)! (log(z/u)) l-1 du = 0 + 2πi (k -2)!(l -1)! z 1 (log u) k-2 .(log(z/u)) l-1 du = 2πi (k -2)!(l -1)! (log z) k+l-2 1 0 v k-2 (1 -v) l-1 dv = 2πi (k -2)!(l -1)! (log z) k+l-2 B(k -1, l) = 2πi (k -2)!(l -1)! (log z) k+l-2 Γ(k -1)Γ(l) Γ(k + l -1) = 2πi (k + l -2)! (log z) k+l-2 = -∆ 1 (Li k+l-1 )
The polylogarithm ring is a good example showing that the eñe ring structure corresponds to the twisted Hadamard structure (see [START_REF] Érez-Marco | The eñe product for a commutative ring[END_REF] Section 10). We note also that the monodromy formula suggests that the proper normalization of the polylogarithm functions is

li k (z) = - 1 2πi Li k (z) so that ∆ 1 li k = 1 (k -1)! (log z) k-1 .
6.2. Application 2: Polynomial logarithmic monodromy class. The simplest case of holomorphic monodromy occurs for a polynomial monodromy. We determine now the minimal Hadamard ring containing polynomial monodromies. As a Corollary of our main formula we have the following Proposition: Proposition 6.2. Let F and G be two holomorphic functions with polynomial monodromies,

∆ α F, ∆ β G ∈ C[z]. Then we have that ∆ γ (F ⊙ G) ∈ C[z] ⊕ C[z] log z .
Proof. We write

∆ α F (z) = n a n z n ∆ β G(z) = m b m z m
then, using the holomorphic monodromy formula, we get

∆ γ (F ⊙ G)(z) = - 1 2πi α,β αβ=γ z/β α ∆ α F (u)∆ β G(z/u) du u = - 1 2πi α,β αβ=γ n,m a n b m z m z/β α u n-m-1 du = - 1 2πi α,β αβ=γ n,m,n =m a n b m z m z n-m-1 -γ n-m-1 β n-m-1 (n -m) - 1 2πi α,β αβ=γ n a n b n z n (log z -log γ)
We have the following elementary Lemma: Proof. The result is obtained by recurrence on the exponent l ≥ 0 and integration by parts.

As a Corollary we obtain the stability of the ring C[z, log z] by the holomorphic monodromy formula, and the same computation as for Proposition 6.2 proves the general Theorem: We have proved:

Proposition 6.6. The P LM class is closed under the Hadamard product, and it is the minimal Hadamard ring containing the subclass of functions with polynomial monodromies.

We can be more precise keeping track of the previous computations. Definition 6.7. Let K ⊂ C be a field such that 2πi ∈ K. The P LM (K) class is composed by holomorphic functions in a neighborhood of 0, having only singularities in the ring K[z, log z], with singularities α ∈ K.

Theorem 6.8. The P LM (K) class is closed under the Hadamard product, and it is the minimal Hadamard ring containing the subclass of functions with polynomial monodromies in K[z] with singularities in K.

We have the same result for the eñe product.

6.3. Application 3: Divisor interpretation of the eñe product. As explained in the introduction, the exponential eñe product linearizes is the exponential form of the eñe product. It is remarkable that the formulas for the eñe product are linearized through the exponential function.

Using the monodromy formulas we can prove directly the divisor interpretation of the eñe product. Thus this gives an alternative definition of the eñe product and its properties. Starting from two meromorphic functions in the plane, holomorphic at 0, normalized such that f (0) = g(0) = 1, we can consider their exponential form: The functions F = log f and G = log g are holomorphic germs at z = 0. Their singularities are located at the zeros of f and g respectively. We have constant monodromies,

∆ α F = 2πi n α ∆ β G = 2πi n β
where n α , resp. n β , is the multiplicity of the zero or pole of f , resp. g (negative multiplicity for poles). We can define the eñe product by the exponential eñe product,

f ⋆ g = exp(F ⋆ e G)
this defines a holomorphic germ near 0, with f ⋆ g(0) = 1. We know that the singularities of f ⋆ g are located at the singularities of F ⋆ e G, i.e. at the points γ = αβ.

The monodromies are totally holomorphic, hence we can use formula (4) and compute the monodromies of the singularities γ, which gives

∆ γ (F ⋆ e G) = 1 2πi α,β αβ=γ (2πi n α )(2πi n β ) = (2πi) α,β αβ=γ n α n β
This proves that γ is a zero or pole for f ⋆ g of multiplicity

n γ = α,β αβ=γ n α n β
hence recovering the definition given in [START_REF] Érez-Marco | The eñe product for a commutative ring[END_REF].

It is instructive to note, in view of the infinite divisor interpretation of the eñe product given in [START_REF] Érez-Marco | Eñe product in the transalgebraic class[END_REF], that isolated essential singularities of f and g do correspond to regular poles of F and G, that are a particular case of regular isolated singularities. If we use the analogue of Borel Theorem for regular isolated singularities, we can see that the divisor eñe product interpretation extends further to singularities of f and g of infinite order, for example of the form f (z) = e 1-e z z-1 . These "higher order divisor" extension are left for future work.

Figure 1 . 1 .

 11 Figure 1. Choreographic monodromy integration contour.

= 1 +

 1 n≥1 a n z n g(z) = exp(G(z)) = 1 + b 1 z + b 2 z 2 + . . . = 1 + n≥1 b n z nand we define the eñe product asf ⋆ g(z) = exp(F ⋆ e G(z)) = 1 + c 1 z + c 2 z 2 + . . . = 1 + n≥1 c n z nthen the coefficient c n is a universal polynomial with integer coefficients on a 1 , a 2 , . . . , a n , b 1 , b 2 , . . . , b n .

  0, hence we have: Corollary 1.7. If F and G have only uniform singularities then F ⋆ e G has only uniform singularities, i.e. ∆ γ (F ⋆ e G) = 0.

2

 2

Figure 2 .

 2 Figure 2. Homotopical deformation of the integration path when z 0 turns around 1.

Lemma 6 . 3 .

 63 Let α ∈ C * , and k, l ≥ 0. We havez α u k (log u) l du ∈ Q[α, log α][z, log z] .

Theorem 6 . 4 .Definition 6 . 5 .

 6465 Let F and G be two holomorphic functions with monodromies,∆ α F, ∆ β G ∈ C[z, log z]. Then we have that ∆ γ (F ⊙ G) ∈ C[z, log z] .This motivates the definition of the polynomial logarithmic monodromy class. The polynomial logarithmic monodromy (or P LM )class is the class of holomorphic functions in a neighborhood of 0 having only singularities with monodromies in the ring C[z, log z].

f

  (z) = exp(F (z)) = 1 + a 1 z + a 2 z 2 + . . . = 1 + n≥1 a n z n g(z) = exp(G(z)) = 1 + b 1 z + b 2 z 2 + . . . = 1 + n≥1 b n z n

where ǫ 1 is a local positive circle loop around u = 1, and we take the limit when this loop converges to u = 1. We have

and the second limit is zero because of the integrability condition, and more precisely because

Therefore, for each path η j we get the following extra residue contributions to the integral (the orientations give the proper signs):

Adding these four contributions, and clearing out the two cancellations, only two residues remains and they give a total residual contribution

We can give to this expression a symmetric form using the following elementary Lemma.

Lemma 5.1. We have

Proof. We consider a local loop γ with winding number 1 with respect to z 0 and use the Residue Theorem and the change of variables v = z 0 /u, du = -z 0 dv/v 2 , γ ′ the image of γ that is a local loop with winding number number with respect to 1,

Finally, the residue total contribution is