Resurgence and highest level’s connection-to-Stokes formulæ for some linear meromorphic differential systems

ANNALES

DE LA FACULTÉ DES SCIENCES

Mathématiques

PASCALREMY

Resurgence and highest level’s connection-to-Stokes formulæ for some linear meromorphic differential systems

Tome XXVI, n^o3 (2017), p. 645-685.

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Resurgence and highest level’s connection-to-Stokes formulæ for some linear meromorphic differential

systems

Pascal Remy⁽¹⁾

ABSTRACT. — In this article, we consider a linear meromorphic dif- ferential system with several levels. We prove that the Borel transforms of its highest level’s reduced formal solutions are summable-resurgent and we give the general form of all their singularities. This one is then precised in restriction to some convenient hypotheses on the geometric configura- tion of singular points. Next, under the same hypotheses, we state exact formulæ to express some highest level’s Stokes multipliers of the initial system in terms of connection constants in the Borel plane, generalizing thus formulæ already displayed by M. Loday-Richaud and the author for systems with a single level. As an illustration, we develop one example.

RÉSUMÉ. — Dans cet article, nous considérons un système différentiel linéaire méromorphe à multiples niveaux. Nous démontrons que les trans- formées de Borel de ses solutions formelles réduites de plus haut niveau sont résurgentes-sommables et nous donnons la forme générale de toutes leurs singularités. Celle-ci est ensuite précisée pour certaines configura- tions géométriques des points singuliers. Pour ces mêmes configurations, nous énonçons également des formules exactes permettant d’exprimer les multiplicateurs de Stokes de plus haut niveau du système initial à l’aide de constantes de connexion dans le plan de Borel, généralisant ainsi les formules déjà données par M. Loday-Richaud et l’auteur pour les systèmes de niveau unique. Ces formules sont illustrées par un exemple.

(*)Reçu le 6 septembre 2015, accepté le 29 juin 2016.

Keywords:Linear differential system, multisummability, Stokes phenomenon, Stokes multipliers, resurgence, singularities, connection constants.

Math. classification:34M03, 34M25, 34M30, 34M35, 34M40.

(1) Lycée Les Pierres Vives, 1 rue des Alouettes, 78420 Carrières-sur-Seine, France

— pascal.remy.maths@gmail.com Article proposé par Lucia DiVizio.

1. Introduction

All along the article, we are given a positive integerrě1 and a linear differential system (in short, a differential system or a system) of dimen- sionně2 with meromorphic coefficients of orderr`1 at the origin 0PC of the form

x^r`1dY

dx “ApxqY , Apxq PMnpCtxuq, Ap0q ‰0. (A) Using a finite algebraic extension x ÞÝÑ x^ν of the variable x with ν P N, ν ě1, and a meromorphic gauge transformationY ÞÝÑTpxqY with a suit- able polynomial matrixTpxqin xand 1{xif needed, we can always assume (see [5]) that system (A) admits for formal fundamental solution at 0 a matrixYrpxqof the form Yrpxq “Fpxqxr ^Le^Qp1{xq and normalized as follows:

pN1q: Frpxq PMnpCrrxssqis a formal power series inxsatisfyingFrpxq “ In`Opx^rq, where In denotes the identity matrix of sizen,

pN2q: the matrixLPMnpCqof exponents of formal monodromy reads in a Jordan formL“

J

à

j“1

pλjIn_j `Jn_jq, whereJ is an integerě2, the eigenvaluesλj satisfy 0ďRepλjq ă1 and where

Jnj “

$

’’

’’

’’

’’

&

’’

’’

’’

’’

%

0 ifnj“1

»

—

—

—

—

— –

0 1 ¨ ¨ ¨ 0 ... . .. . .. ... ... . .. 1 0 ¨ ¨ ¨ ¨ ¨ ¨ 0 fi ffi ffi ffi ffi ffi fl

ifnjě2

is an irreductible Jordan block of sizenj, pN3q: Q

ˆ1 x

˙

“

J

à

j“1

qj

ˆ1 x

˙

In_j is a diagonal matrix of polynomials q_jp1{xqin 1{xof degreeďrand without constant term which com- mutes withL.

Recall that normalizations pN1q ´ pN2q guarantee the unicity of Fpxqr as formal series solutions of the homological system associated with system (A) (cf. [5]).

The effective calculation of the Stokes multipliers of Frpxq (= the non- trivial entries of theStokes–Ramis matrices associated withYrpxq, see Def- inition 5.1) is crucial in a large number of theoretical and practical prob- lems (calculation of differential Galois groups [24, 25], integrability of some

Hamiltonian systems [26, 27], etc.). Thereby, in the last decades of the twen- tieth century, several approaches, issuing from the summability and multi- summability theories and essentially based on integral methods such that Cauchy–Heine integral and Laplace transformations, were given by many authors under more or less generic assumptions on system (A) (see for in- stance [2, 4, 6, 7, 8, 9, 11, 12, 15, 20]).

More recently, in a 2011 article [19], M. Loday-Richaud and the author combined, in the case where system (A) has the uniquelevel 1 (see Defini- tion 2.3 for the exact definition of levels), this “summation” approach with the “resurgence” approach due to J. Écalle. Doing that, they derived, from a full description of the resurgent structure of the Borel transform Fp ofFr, explicit formulæ relating the Stokes multipliers ofFrtoconnection constants given by some analytic continuations of Fp at its various singular points, providing thus a new efficient tool for the effective calculation of the Stokes multipliers ofFr.

Afterwards, these so-called connection-to-Stokes formulæ were general- ized by the author to systems with an arbitrary single levelr[32] and to the lowest level of systems with multi-levels [31] by respectively replacing, via the classical method of rank reduction, the initial system by its r-reduced and its lowest level’s reduced system. One knows indeed perfectly relate the Stokes–Ramis matrices of the initial system with those of its reduced sys- tem [17].

In the present article, we assume that system (A) has multi-levels, say r1 ă ¨ ¨ ¨ ă rp with pě 2, and we propose to extend these connection-to- Stokes formulæ to the highest level rp. To do that, the organization of the paper is as follows. In Section 2, we first recall some definitions and ba- sic properties about levels and about therp-reduced system (A) associated with system (A). In Section 3, we describe the complete resurgent structure of the Borel transforms of the formal solutions of system (A). In particular, we show that these functions are summable-resurgent (Theorem 3.3) and we give the general form of all their singularities (Theorem 3.6). These two the- orems are then proved in Section 4 by reducing system (A) into a convenient scalar linear differential equation with polynomial coefficients and by apply- ing a method similar to the one of [32]. In Section 5, we restrict our study to the case where the Borel transforms we consider have “good” singularities.

In this case, we define their connection constants at their various singular points and we relate these ones to the highest level’s Stokes multipliers ofFr throughout explicithighest level’s connection-to-Stokes formulæ, generaliz- ing thus formulæ already obtained in [19, 31, 32] . We illustrate this result with a numerical example.

2. Preliminaries

In this section, we recall some definitions we are needed in the sequel and we introduce the highest level’s reduced system associated with system (A) which will play a central role all along the article.

2.1. Levels, Stokes values and anti-Stokes directions

Split the matrixFrpxq into J column-blocks fitting to the Jordan block structure ofL(for`“1, . . . , J, the matrixFr<sup>‚;`</sup>pxqhasn` columns):

Frpxq “”

Fr^‚;1pxq Fr^‚;2pxq ¨ ¨ ¨ Fr^‚;Jpxq ı

.

Definition 2.1. — Letj, `P t1, . . . , Jube such thatq<sub>j</sub>ıq<sub>`</sub>. We denote pqj´q`q

ˆ1 x

˙

“ ´αj,`

x<sup>rj,`</sup> `o ˆ 1

x^rj,`

˙

withα_{j,`</sub>‰0 andr<sub>j,`} PN^˚“Nzt0u. Then,

‚ the degreer<sub>j,`</sub> is called a level ofFr<sup>‚;`</sup>pxq,

‚ the coefficientαj,` is called a Stokes value of levelrj,` ofFr^‚;`pxq,

‚ the directions of maximal decay of e<sup>pqj´q`qp1{xq</sup>, i.e. the rj,` direc- tions argpαj,`q{rj,` modp2π{rj,`q along which ´αj,`{x<sup>rj,`</sup> is real negative, are called anti-Stokes directions of levelrj,` ofFr^‚;`pxq.

Note that a Stokes value (resp. an anti-Stokes direction) ofFr^‚;`pxq may be with several levels. Note also that the term “anti-Stokes direction” is not universal; sometimes, one calls such a direction “Stokes direction”.

Notation 2.2. — For all`P t1, . . . , Ju, we denote byR`:“ tρ`;1ă ¨ ¨ ¨ ă ρ<sub>`;p`</sub>uwithp<sub>`</sub>ě1 the set of all the levels ofFr^‚;`pxq.

Note that, according to normalizationpN₃q, all the levelsρ<sub>`;k</sub> of all the Fr<sup>‚;`</sup>pxqare integer; one refers sometimes this case as theunramified case.

Note also that, for all `, we have ρ`;p_` ď r the rank of system (A).

Actually, if there exists `0 such that ρ`₀;p<sub>`0</sub> ă r, then ρ`;p_{`</sub> ă r for all ` and polynomialsqj have the same degree r and the same terms of highest degree. One then reduces to the case ρ`;p<sub>`} “ r by means of a change of unknown vector of the form Y “ Ze^qp1{xq with a convenient polynomial qp1{xq Px^´1Crx^´1s. Recall that such a change does not affect levels, Stokes values, anti-Stokes directions nor Stokes–Ramis matrices (see Section 5.1 for their exact definition) of system (A).

Definition 2.3. — One calls

‚ level ofFpxqr (or of system (A))any level of the Fr^‚;`pxq’s,

‚ Stokes value of Frpxq (or of system (A)) any Stokes value of the Fr^‚;`pxq’s,

‚ anti-Stokes direction of Fpxqr (or of system (A)) any anti-Stokes direction of theFr^‚;`pxq’s.

Notation 2.4. — We denote byR:“ tr₁ă ¨ ¨ ¨ ăr_puwithpě1 the set of all the levels ofFrpxq(or of system (A)). We haveR“R1Y ¨ ¨ ¨ YRJ and rp“rthe rank of system (A).

Since the case p “ 1 was already investigated in great details in [19]

(case r“1) and [32] (caser ě2), we suppose from now onpě2, that is system (A) has at least two levels. Note however that some column-blocks Fr<sup>‚;`</sup>pxqmay have the unique levelr, i.e.p`“1 andR`“ tru.

2.2. Highest level’s reduced system

The highest level’s reduced system (= r-reduced system) associated with system (A) is the unique system of the variable t “ x^r having mero- morphic coefficients at 0 P C and the formal solutions rYrpxq, x^´1Yrpxq, . . . , x^´pr´1qYrpxqs for a given choicex“t^1{r of ar-th root oft[17]. Such a choice being made, we denote from now onµ:“e^´2iπ{r. Then, ther-reduced system of system (A) reads as

rt²dY

dt “AptqY (A)

withAptq PMrnpCttuqthernˆrn-analytic matrix defined by

Aptq “

»

—

—

—

—

—

—

— –

A^r0sptq tA^rr´1sptq ¨ ¨ ¨ ¨ ¨ ¨ tA^r1sptq

A^r1sptq A^r0sptq . .. ... ... . .. . .. . .. ... ... . .. A^r0sptq tA^rr´1sptq A^rr´1sptq ¨ ¨ ¨ ¨ ¨ ¨ A^r1sptq A^r0sptq

fi ffi ffi ffi ffi ffi ffi ffi fl

´

r´1

à

u“0

utIn

where the A^rusptq P MnpCttuq are the r-reduced series of Apxq uniquely determined by the relation

Apxq “A^r0spx^rq `xA<sup>r1s</sup>px<sup>r</sup>q ` ¨ ¨ ¨ `x^r´1A^rr´1spx^rq.

By construction, system (A) has levelsď1. Moreover, it admits as formal fundamental solution at 0 the matrix

ĂYptq “

»

—

—

—

— –

Yrpt^1{rq Yrpµt^1{rq ¨ ¨ ¨ Yrpµ^r´1t^1{rq pt^1{rq^´1Yrpt^1{rq pµt^1{rq^´1Yrpµt^1{rq ¨ ¨ ¨ pµ^r´1t^1{rq^´1Yrpµ^r´1t^1{rq

.. .

.. .

.. .

pt^1{rq^´pr´1qYrpt^1{rq pµt^1{rq^´pr´1qYrpµt^1{rq ¨ ¨ ¨ pµ^r´1t^1{rq^´pr´1qYrpµ^r´1t^1{rq fi ffi ffi ffi ffi fl .

This one reads more precisely on the formYrptq “FrptqYr₀ptqwith

‚ ^{Fptq “r}

»

—

—

—

—

—

—

—

—

—

— –

Fr^r0sptq trF^rr´1sptq ¨ ¨ ¨ ¨ ¨ ¨ tFr^r1sptq

Fr^r1sptq Fr^r0sptq . ..

.. . ..

. . .. . .. . ..

.. . ..

. . .. Fr^r0sptq trF^rr´1sptq Fr^rr´1sptq ¨ ¨ ¨ ¨ ¨ ¨ Fr^r1sptq Fr^r0sptq

fi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi fl

and

‚ ^ĂY0ptq “

»

—

—

—

—

— –

ptr¹q^Λ0e^Q0ptq pµt¹rq^Λ0e^Q1ptq ¨ ¨ ¨ pµ^r´1t¹rq^Λ0e^Qr´1ptq ptr¹q^Λ1e^Q0ptq pµt¹rq^Λ1e^Q1ptq ¨ ¨ ¨ pµ^r´1t¹rq^Λ1e^Qr´1ptq

.. .

..

. . ..

.. .

pt¹rq^Λr´1e^Q0ptq pµt¹rq^Λr´1e^Q1ptq ¨ ¨ ¨ pµ^r´1t¹rq^Λr´1e^Qr´1ptq fi ffi ffi ffi ffi ffi fl

whereQkptq “Qp1{pµ^kt^1{rqqand Λk :“L´kIn for allk“0, . . . , r´1. The formal seriesFr^rusptq P MnpCrrtssq are ther-reduced series of Frpxq and are defined in the same way as theA^rusptq’s. In particular, the initial condition Fpxq “r In`Opx<sup>r</sup>q(see normalizationpN<sub>1</sub>q) impliesFrptq “Irn`Optq.

Let us now splitFrptqintorcolumn-blocksFr^‚;vptqof sizernˆn:

Frptq “”

Fr^‚;1ptq Fr^‚;2ptq ¨ ¨ ¨ Fr^‚;rptq ı

,

then, each Fr^‚;vptq into J column-blocks Fr^{‚;v,`</sup>ptq as Frpxq (the matrix Fr<sup>‚;v,`}ptqhas sizernˆn`):

Fr^‚;vptq “

”

Fr^‚;v,1ptq Fr^‚;v,2ptq ¨ ¨ ¨ Fr^‚;v,Jptq ı

.

LetθPR{2πZbe a non anti-Stokes direction of system (A) andθ:“rθ.

Due to the classical theory of multisummability of linear meromorphic sys- tems (see for instance [2, 3, 11, 16, 17, 18, 23, 25]), the formal seriesFr<sup>‚;v,`</sup>ptq areρ<sub>`</sub>-summable for allv“1, . . . , r in directionθ withρ_{`</sub> :“ pρ<sub>`;1}ă ¨ ¨ ¨ ă ρ_`;p

` “1qand ρ<sub>`;k</sub> :“ρ_`;k{r. In particular, applying Balser–Tougeron the- orem [1] (see also [18, Thm. 7.4.5]), we get the following proposition which

provides us a first result about the formal Borel transform⁽¹⁾ Fp^{‚;v,`</sup>pτq of Fr<sup>‚;v,`}ptq.

Proposition 2.5. — Let vP t1, . . . , ruand`P t1, . . . , Ju. Then,

‚ Casep<sub>`</sub>“1:Fp<sup>‚;v,`</sup>pτqdefines an analytic function on a disc centered at the origin0PC,

‚ Case p<sub>`</sub> ě 2: Fp<sup>‚;v,`</sup>pτq is ρ¹_{`</sub>-summable in direction θ with ρ<sup>1</sup><sub>`} :“

pρ¹_{`;1</sub> ă ¨ ¨ ¨ ăρ<sup>1</sup><sub>`;p}

`´1q andρ<sup>1</sup><sub>`;k</sub> :“ρ_{`;k</sub>{p1´ρ<sub>`;k}q “ρ_{`;k</sub>{pr´ρ<sub>`;k}q;

moreover, its sum defines an analytic function on a sector with ver- tex0, bisected byθ and opening larger than π{ρ¹_`;p

`´1.

We denote below byFp^{‚;v,`</sup><sub>θ</sub> pτqthe function thus defined and by Σ<sup>v,`}_θ its cor- responding domain of analyticity.

In Section 3, we propose to investigate, for anyv and `, the resurgent structure ofFp<sup>‚;v,`</sup>_θ , that is, the analytic continuations of Fp^{‚;v,`</sup><sub>θ</sub> outside the domain Σ<sup>v,`}_θ . In particular, we prove that Fp^‚;v,`_θ is summable-resurgent⁽²⁾ (Theorem 3.3) and we give the general form of all its singularities (Theo- rem 3.6), generalizing thus the results already obtained by M. Loday-Richaud and the author in [19, 32] for systems with a single level.

3. Resurgence and singularities

In this section, we just state the results allowing to describe the resurgent structure of functionsFp^‚;v,`_θ above. These ones will be proved later in Sec- tion 4. In the sequel, we fixθPR{2πZa non anti-Stokes direction of initial system (A) and we set θ“rθ as before.

3.1. Summable-resurgence theorem

Recall that aresurgent function is an analytic function near the origin which can be analytically continued on all a convenient Riemann surface.

More precisely, one has the following.

Definition 3.1 (Resurgent function). — Let Ω ĂC be a finite subset ofCcontaining0. A function defined and analytic near0 is said to be

(1)Recall that the formal Borel transform of a formal seriesř

mě0amt^m PCrrtssis defined bya0δ`ř

mě1amτ^m´1

pm´1q!, whereδdenotes the Dirac distribution at 0.

(2)See Definition 3.2.

‚ resurgent with singular support Ω,0when it can be analytically con- tinued on the whole Riemann surface RΩ defined as (the terminal end of) all homotopy classes inCzΩof paths issuing from0and by- passing all points ofΩ(only homotopically trivial paths are allowed to turn back to0); in particular, such a function is analytic at0 in the first sheet,

‚ resurgent with singular support Ω,r0when it can be analytically con- tinued on the whole Riemann surfaceRr_Ω:“the universal cover of CzΩ.

We denote byResΩ,0andRes_Ω,r0the sets of resurgent functions with singular supportΩ,0 and of resurgent functions with singular supportΩ,r0.

Recall that the difference betweenR_ΩandRr_Ωjust lies in the fact thatR_Ω has no branch point at 0 in the first sheet. In particular, we have a natural injection ResΩ,0 ãÑ Res_Ω,r0. Recall also that the choice of the Riemann surface RrΩ or RΩ only depends on the fact that the function we consider has a singular point or not at the origin 0PC(i.e. in the first sheet).

Definition 3.2 (Summable-resurgent function). — A resurgent func- tion of ResΩ,0 (resp. Res_Ω,r0) is said to be summable-resurgent if it grows at most exponentially on any bounded sector of infinity ofRΩ (resp. RrΩ).

We denote by Res^sum_Ω,0 (resp. Res^sum

Ω,r0 ) the set of summable-resurgent func- tions with singular support Ω,0 (resp. Ω,r0). As before, we have a natural injectionRes^sum_Ω,0 ãÑRes^sum

Ω,r0 .

We are now able to state the main result of this section.

Theorem 3.3 (Summable-resurgence theorem). — Let v P t1, . . . , ru and ` P t1, . . . , Ju. Let p` ě1 be the number of levels of Fr<sup>‚;`</sup>pxq. Let Ω<sup>˚</sup><sub>`</sub> be the set of Stokes values of level r of Fr<sup>‚;`</sup>pxq (see Definition 2.1) and Ω`:“Ω^˚_` Y t0u. Then,

‚ Casep<sub>`</sub>“1:Fp<sup>‚;v,`</sup>_θ pτq PRes^sum_Ω

`,0,

‚ Casep<sub>`</sub>ě2:Fp<sup>‚;v,`</sup>_θ pτq PRes^sum_Ω

`,r0.

In particular, Theorem 3.3 tells us that the only possible singular points of Fp^{‚;v,`</sup><sub>θ</sub> pτq are 0 and the Stokes values of level r of Fr<sup>‚;`}pxq. The general form of the singularities ofFp^‚;v,`_θ pτqat these various points is precised just below.

3.2. General form of singularities

Before stating the structure of the singularities of the Fp^‚;v,`_θ pτq’s, let us recall some definitions and notations about the singularities. For more details, we refer for instance to [14, 21, 22, 33].

3.2.1. Some spaces of singularities

Denote byOthe space of holomorphic germs at 0PCand byOrthe space of holomorphic germs at 0 on the Riemann surfaceCr of the logarithm. One calls any element of the quotient spaceC :“O{Or asingularity at 0. Recall thatCis also denoted bySING0by J. Écalle et al. (cf. [33] for instance). Recall also that the elements ofC are called micro-functionsby B. Malgrange [21, 22] by analogy with hyper- and micro-functions defined by Sato, Kawai and Kashiwara in higher dimensions.

The elements ofCare usually denoted with a nabla, like

∇

h, for a singu- larity of the functionh. A representative of

∇

hinOris often denoted byqhand is calleda major ofh.

It is worth to consider the two natural maps

can :Or ÝÑC“O{Or the canonical map and var : C ÝÑOr the variation map, action of a positive turn around 0 defined by var

∇

h “ qhpτq ´qhpτ e^´2iπq, where qhpτ e^´2iπq is the analytic continuation ofqhpτq along a path turning once clockwise around 0 and close enough to 0 forqhto be defined all along (the result is independent of the choice of the majorqh). The germ var

∇

h is calledthe minor of

∇

h.

One can not multiply two elements of C, but an element of C and an element ofO:α

∇

h:“canpαqhq “

∇

pαhqfor allαPO and

∇

hPC . On the other hand, one can define a convolution productf on C by setting

∇

h₁f

∇

h₂ :“

canpqh₁˚uqh₂q, whereqh₁˚uqh₂is the truncated convolution product pqh1˚uqh2qpτq:“

żτ´u u

qh1pτ´ηqqh2pηqdηPOr

withuarbitrarily close to 0 satisfyingτP s0, urand argpτ´uq “argpτq ´π.

Note that

∇

h1f

∇

h2 makes sense since it does not depend on u, nor on the

choice of the majorsqh₁andqh₂. The convolution productfis commutative and associative onC with unitδ:“can` ₁

2iπτ

˘. The action of _dτ^d onC is defined by

d dτ

∇

h“δ¹f

∇

h and satisfies relation

d dτ

ˆ_∇ h1f

∇

h2

˙

“ ˆ d

dτ

∇

h1

˙ f

∇

h2“

∇

h1f ˆ d

dτ

∇

h2

˙ .

Finally, the multiplication byτ is anf-derivation, i.e.

τ ˆ_∇

h1f

∇

h2

˙

“ ˆ

τ

∇

h1

˙ f

∇

h2`

∇

h1f ˆ

τ

∇

h2

˙ .

In the sequel of this article, we shall use especially the following classical subspaces ofC.

‚ The subspaceC^ď1 of singularities for which the variation defines an entire function on allCrwith exponential growth of orderď1 on any bounded sector of infinity⁽³⁾. Recall that this space is isomorphic to the space of analytic functions with subexponential growth at 0PCr[14, pp. 46-48]. In particular, any powert^λand any exponential e^Ppt1{pq with p ě2 and Pptqpolynomial in t of degree ăp define singularities inC^ď1.

‚ The subspace

∇

Nil^s-res_Ω,0 (resp.

∇

Det^s-res_Ω,r

0 ) of singularities for which there exists a major of the form

ÿ

finite

hα,ppτqτ^αplnτq^p

with αPC, pPN and hα,ppτq P Res^sum_Ω,0 (resp. hα,ppτq P Res^sum

Ω,r0

holomorphic on a punctured disc at 0 in the first sheet).

Definition 3.4. — The elements of

∇

Nil^s-res_Ω,0 (resp.

∇

Det^s-res

Ω,r0 ) are called summable-resurgent singularities of Nilsson class with singular support Ω,0 (resp.of finite determination with singular support Ω,r0).

For anyωPC^˚, we denote byC_|ωthe space of singularities atω, i.e. the spaceCtranslated from 0 toω. A functionqhis then a major of a singularity atω ifqhpω`τqis a major of a singularity at 0. In the same way, we define the translated spaceC^ď1_|ω, etc. . .

(3)or, equivalently, for which there exists a major satisfying this same property [14].

3.2.2. Description of singularities

GivenvP t1, . . . , ruand`P t1, . . . , Ju, the behavior ofFp<sup>‚;v,`</sup>_θ at a singu- lar pointω PΩ^˚<sub>`</sub> depends, of course, on the “homotopic class” of the path γof analytic continuation followed from any point a‰0 of Σ<sup>v,`</sup>_θ to a neigh- borhood of ω. Note in particular that “homotopic class” implies that the behavior ofFp^‚;v,`_θ does not depend on the choice ofa. We denote below by

‚ Fp^{‚;v,`</sup><sub>θ;ω,γ</sub> the analytic continuation ofFp<sup>‚;v,`}_θ along the path γ,

‚

∇

F^{‚;v,`</sup><sub>θ;ω,γ</sub> :“ canpFp<sup>‚;v,`}_θ;ω,γq the singularity of Fp^{‚;v,`</sup><sub>θ</sub> at ω defined by Fp<sup>‚;v,`}_θ;ω,γ.

Let us now introduce the key notion of front of a singularity [31, 32] . Definition 3.5. — Let ` P t1, . . . , Ju and ω P Ω<sup>˚</sup><sub>`</sub> a Stokes value of levelrof Fr<sup>‚;`</sup>pxq. We call front of levelrofω the set of all the polynomials pqj ´q`qp1{xq with leading terms ´ω{x^r. We denote it by F r`pωq and we have

F r`pωq:“

"

´ω

x^r `q`,ω;k

ˆ1 x

˙

; k“1, . . . , s`,ω

* ,

where s`,ω is an integer ě 1 and where all the q`,ω;kp1{xq are polynomials in 1{x of degree ă r and without constant term. Moreover, ω (hence, its corresponding singularity

∇

F<sup>‚;v,`</sup><sub>θ;ω,γ</sub> too) is said to be of a good front when s`,ω“1 and with a bad frontotherwise.

In the special case where ω has a good front, we simply denote q_{`,ω</sub> for q<sub>`,ω;1}. Then,

F r`pωq “

"

´ω x^r `q`,ω

ˆ1 x

˙*

and we more precisely say that ω (and its corresponding singularity too) has a good monomial front when q`,ω ” 0 and a good nonmonomial front otherwise.

We are now able to state the main result of this section.

Theorem 3.6 (General form of

∇

F^{‚;v,`</sup><sub>θ;ω,γ</sub>). — Let v P t1, . . . , ru and ` P t1, . . . , Ju. Let ω PΩ^˚<sub>`</sub> be a Stokes value of level r of Fr<sup>‚;`}pxqand γ a path onCzΩ` starting from a point ofΣ<sup>v,`</sup>_θ and ending in a neighborhood of ω.

(1) Suppose thatω has a good front. Let Q`,ω“

"

q`,ω

ˆ 1

µ^v´1t^1{r

˙

; v“1, . . . , r

*

withµ“e^´2iπ{r. Then,

∇

F^‚;v,`_θ;ω,γP ÿ

qPQ_`,ω

∇

Nil^s-res_Ω`´ω,0f^∇e^q_|ω.

In particular, ifω has moreover a monomial front, then

∇

F^‚;v,`_θ;ω,γP

∇

Nil^s-res_Ω`´ω,0|ω.

(2) Suppose thatω has a bad front. Let Q`,ω“

"

q`,ω;k

ˆ 1

µ^v´1t^1{r

˙

; k“1, . . . , s`,ω andv“1, . . . , r

*

withµ“e^´2iπ{r. Then

∇

F^‚;v,`_θ;ω,γP ÿ

qPQ_`,ω

∇

Det^s-res_Ω

`´ω,r0f^∇e^q_|ω.

Notation^∇e^qstands for the singularity ofC^ď1defined bye^q(see Section 3.2.1).

A more precise description of singularities with good monomial front will be given later in Section 5. For the moment, let us prove our two main Theorems 3.3 and 3.6.

4. Proofs of Theorems 3.3 and 3.6

Before starting the proofs, let us first begin by some reminders about the Borel transformation which shall play a central role.

4.1. Borel transformation

Definition and properties.LetαPR{2πZand let hptq be a function defined and holomorphic on a domain containing a sector with vertex 0, bisected by α and opening larger than π. Under “good” hypotheses on h at 0 [24], the Borel transform (of level 1) ofhin directionαis given by the integral

Bαphptqqpτq “phαpτq:“ 1 2iπ

ż

Γ_α

hptqe^´t{τdt t²,

where Γαdenotes the image bytÞÑ1{tof a Hankel contour directed by the directionαand oriented positively⁽⁴⁾. Using Hankel’s formula for the inverse of the Gamma function, we obtainBαpt^λqpτq “τ^λ´1{Γpλqfor allλPCzp´Nq andαPR{2πZ; whenhptq “t^´mwithmě1, we get a natural generalization Bαpt^´mq “δ^pmq the m-th derivative of the Dirac distribution at 0 (hence, the coherence with the definition of the formal Borel transformation, see footnote 1).

The Borel transformationBα changes the derivationt²_dt^d into the multi- plication byτand the multiplication by 1{tinto the derivation _dτ^d . In partic- ular, it changes derivation _dt^dkk into _dτ<sup>dk`1</sup>k`1pτ^{k d}_dτ^k´1k´1qfor anykě1. Moreover, it changes the ordinary product¨into the convolution product˚:

hptqkptq ÞÝÑphα˚pkαpτq:“

żτ 0

phαpτ´ηqpkαpηqdη

when phα and pkα are both integrable at 0 (note that δ is the unit of ˚).

Finally, it changes the multiplication bye^´ω{tinto the translation by ω.

Some classes of functions.Among all the classes of functions on which one can apply the Borel transformation, we shall actually use in the sequel of the article only those of one of the following two types:

‚ hptq P Or has a subexponential growth at the origin (we denote below hptq POr^ďexp), that is, for all εą0, there exists a constant Cε ą 0 such that |hptq| ď Cεe^ε{|t| (or, what amounts the same, lim sup

|t|Ñ0

p|t|lnp|hptq|qq “0) uniformly on any bounded sector of the formθ₁ăargptq ăθ₂. In particular, any analytic functionhptq PO at 0, any power t^λ of t, any power plntq^m of the logarithm and any exponentiale^Ppt´1{pqwith a polynomialPptqin tof degreeăp belong toOr^ďexp,

‚ hptqis thek-sum of ak-summable seriesrhptq PCrrtss in directionα withk:“ pk1ă ¨ ¨ ¨ ăks“1qandsě1.

For the first one, the existence ofph_α is straightaway from the property ofh at 0 and one can show thatphαdefines in this case an entire function on allCr (= the Riemann surface of the logarithm) with exponential growth of order ď1 on any bounded sector at infinity. We denote belowph_αpτq PO^ď1prCq.

In the special case wherehptq PO is analytic at 0, one has more precisely ph_αpτq P O^ď1pCq. As for the second one, the existence of ph_α is due to the

(4)Observe that we need a contour that ends at 0 since the functions we consider are studied near the origin; if we worked at infinity, we would use a Hankel contour itself.

continuity of h at 0; furthermore, one can check that ph_α coincides with the function defined from the Balser–Tougeron theorem by the formal Borel transform ofrh. In particular, denoting by F<sup>‚;v,`</sup><sub>θ</sub> ptqtheρ<sub>`</sub>-sum of Fr^{‚;v,`</sup>ptq in directionθ (see p. 650), we haveB<sub>θ</sub>pF<sup>‚;v,`}_θ q “Fp^‚;v,`_θ .

Extended Borel transformation.Since the Borel transformBαphptqq of any hptq P Or^ďexp may be integrable, or not, at 0 (see for instance the Borel transform of t^λ just above), the convolution product ˚ may not be defined, as well as the Laplace transform of such functions. To circumvent this problem, the idea consists then in consideringBαphptqqnot as a function, but as a singularity whose the variation isphαpτq. More precisely, one has the following.

Proposition 4.1(Écalle, [14, pp. 46–48]). — LetαPR{2πZbe a direc- tion. Then, the Borel transformationBαcan be extended to an isomorphism

B^ext_α : ˆ

Or^ďexp,`,¨, t²d dt

˙ ÝÑ`

C^ď1,`,f, τ¨˘

, B_α^extphq “

∇

hα

of C-differential algebras⁽⁵⁾ so that varp

∇

hαq “ phα for all h P Or^ďexp. Its inverse is the Laplace transformationL^ext_α defined as follows: given

∇

hPC^ď1, qha major of

∇

handph“varp

∇

hq , L^ext_α p

∇

hqptq:“

ż

γα,ε

qhpτqe^´τ{tdτ` ż8e^iα

εe^iα

phpτqe^´τ{tdτ,

whereγα,ε denotes a circle centered at the origin and going fromεe^ipα´2πq toεe^iα,εą0 small enough.

Note thatL^ext_α p

∇

hqmakes sense since it does not depend on the choice of εnor on the chosen majorqh; in particular, for a choiceqhPO^ď1pCqr , one has

L^ext_α p

∇

hqptq:“

ż

γ_α

qhpτqe^´τ{tdτ,

whereγα denotes a Hankel path directed by directionαand oriented posi- tively. Note also that, ifphis integrable at 0, thenL^ext_α p

∇

hqcoincides with the

“classical” Laplace transform L_αpphqptq:“

ż8e^iα 0

phpτqe^´t{τdτ.

(5)The stability ofC^ď1 under the convolution productfis due to the fact that, for any singularity

∇

hPC^ď1, one can always choose a majorqhinO^ď1pCrq(see footnote 3).

The following relations are essentially known:

∇

pt^λplntq^pq_α“can ˆ d^p

dλ^p

ˆ τ^λ´1 p1´e^´2iπλqΓpλq

˙˙

,

∇

pt^mq_α“can

ˆτ^m´1lnτ pm´1q!

˙

, ^∇1_α“δ,

∇

pt^´mq_α“δ^pmq

for all λ P CzZ, p P N, m P N^˚ and α P R{2πZ. More generally, let Crt^λ,plntq^psλPC,pPN ĂOr^ďexp denote the space of C-linear combinations of terms of the formt^λplntq^p with λPC andpPN. Let

∇

Cαrt^λ,plntq^psλPC,pPN

be its image byB^ext_α . Then, for any

∇

hP

∇

Cαrt^λ,plntq^psλPC,pPN, there exists a majorqhin Crτ^µ,plnτq^qsµPC,qPN. The following result will be useful later.

Proposition 4.2. — Let Ω Ă C be a finite subset of C containing 0.

Let p ě 2 and let qp1{xq be a polynomial in 1{x of degree ăp. Then, the four spaces

∇

Nil^s-res_Ω,0 ,

∇

Det^s-res

Ω,r0 ,

∇

Nil^s-res_Ω,0 f^∇e^qpt´1{pq and

∇

Det^s-res

Ω,r0 f^∇e^qpt´1{pq are stable under derivation _dτ^d and under f-convolution by an element of

∇

Cαrt^λ,plntq^psλPC,pPN . They are also stable by multiplication byτ.

Proof. — We just prove the stability of

∇

Nil^s-res_Ω,0 f^∇e^qpt´1{pqand

∇

Det^s-res

Ω,r0 f

∇e^qpt´1{pqby the multiplication byτ. The other stabilities are straightforward and are left to the reader. Let

∇

h P

∇

Nil^s-res_Ω,0 (resp.

∇

Det^s-res

Ω,r0 ). Since the multiplication byτ is af-convolution, we have

τ ˆ_∇

hf^∇e^qpt´1{pq

˙

“ ˆ

τ

∇

h

˙

f^∇e^qpt´1{pq`

∇

hf

´

τ^∇e^qpt´1{pq

¯ ,

where, due to the properties of the Borel transformationB^ext_α , τ^∇e^qpt´1{pq“B_α^ext

ˆ t²d

dte^qpt´1{pq

˙

“

∇

Pf^∇e^qpt´1{pq

with

Pptq “ ´1

pt^pp´1q{pdq dt

´ t^´1{p

¯

PtCrt^´1{ps.

Indeed,e^qpt´1{pqPOr^ďexp(note that the choice of the directionαis arbitrary).

Consequently, τ

ˆ_∇

hf^∇e^qpt´1{pq

˙

“ ˆ

τ

∇

h`

∇

hf

∇

P

˙

f^∇e^qpt´1{pq

withτ

∇

h`

∇

hf

∇

P P

∇

Nil^s-res_Ω,0 (resp.

∇

Det^s-res

Ω,r0 ), which completes the proof.

Let us now consider ak-summable seriesrhptq PCrrtssin directionαwith k“ pk₁ă ¨ ¨ ¨ ăk_s“1qandsě1. As before, we denote byh_αptqitsk-sum in directionαand byph_αpτqthe Borel transform ofh_αin directionα. Assume also thatph_αpτq PRes^sum_Ω,0 ifs“1 andph_αpτq PRes^sum

Ω,r0 ifsě2. Then, the extended Borel transformationB_α^extof Proposition 4.1 above can be applied tohαand one thereby defines a unique singularity

∇

hαsatisfying varp

∇

hαq “phα

andL^ext_α p

∇

hαq “hα. For example, fors“1 andrhptq “a`rcptq PC‘tCrrtss, one has

phαpτq “aδ`pcαpτq PCδ‘Res^sum_Ω,0 and

∇

hα“can ˆ a

2iπτ `pcαpτqlnτ 2iπ

˙ . In particular,

∇

h_αP

∇

Nil^s-res_Ω,0 . More generally, one has the following classical result.

Proposition 4.3. — With conditions as above.

(1) Cases“1.

(a) Let λPCandpPN. Then,

∇

phαptqt^λplntq^pq_αP

∇

Nil^s-res_Ω,0 . (b) Conversely, let

∇

h “ canphpτqτ^λplnτq^pq P

∇

Nil^s-res_Ω,0 and write hpτqin a neighborhood of 0PCas

hpτq “ ÿ

mě0

hmτ^m.

Then, for any directionαPR{2πZsuch thatΩXs0,8e^iαr“ H, L^ext_α p

∇

hq “

p

ÿ

k“0

ˆp k

˙

h_λ,p´k;αptqt^λ`1plntq^k,

where, for all `“0, . . . , p,hλ,`;αptqis the1-sum in directionα of the formal series

rh_λ,`ptq “2iπ ÿ

mě0

d^{`</sup> dz<sup>`}

ˆ e^´iπz Γp1´zq

˙

|z“m`1`λ

h_mt^m.

Moreover, the Borel transform phλ,`;αpτq in direction α of h<sub>λ,`;α</sub>ptqbelongs toRes^sum_Ω,0 .

(2) Case sě2.Let λPC andpPN. Suppose also that phα is of finite determination at0. Then