Stokes versus resurgence multipliers for ordinary linear differential ...

Stokes versus resurgence multipliers for ordinary linear
di¤erential systems with a single arbitrary level
Pascal REMY
October 2, 2008
Abstract
We provide a proof of the summability-resurgence of solutions of any ordinary linear
di¤erential system with a single arbitrary level r following Ecalle’s method by regular perturbation
and majorant series. We deduce from it a precise description of singularities in the
Borel plane with an application to the exact correspondence between resurgence multipliers
(or connection constants) in the Borel plane and Stokes multipliers in the Laplace plane.
0 Introduction
The main two “e¤ective”approaches to the meromorphic classi…cation of ordinary linear di¤erential
systems at 0 are, on one hand, the theory of summation (r-summation or multi-summation) — a
full set of meromorphic invariants is then given by the Stokes matrices of the system— and, on the
other hand, the theory of resurgence, a full set of invariants being then given by alien derivatives.
The Stokes matrices here considered are the transition matrices between asymptotic solutions on
each side of the singular summation directions (anti-Stokes directions) in the space C of the initial
variable x, also called the Laplace plane. The alien derivatives are de…ned as an average of various
analytic continuations of the Borel transform of solutions around their various singularities in the
so-called Borel plane. In view to their numerical calculation one has to determine the connection
constants (or, resurgence multipliers) linking analytic continuations of the Borel transform
of solutions to a given fundamental solution at the various singularities. Numerically, it appears
easier and of lower cost to calculate the resurgence multipliers associated with one speci…c analytic
continuation than to calculate an average of several analytic continuations. However, while Stokes
matrices and alien derivatives are connected via a matrix-log formula, to come back to Stokes
matrices from resurgence multipliers one needs to make explicit the connection between the two.
In this paper, we are given an ordinary linear di¤erential system (in short, a di¤erential system
or a system) of dimension n 2 with analytic coe¢ cients at 0 in C and rank r 1
r+1 dY
(1) x
dx
= A(x)Y ; A(x) 2 Mn(Cfxg)
1

together with a formal fundamental solution
eY (x) = e F (x)x L e Q(1=x)
that we assume to be prepared as follows:
eF (x) = In + X
Fmx m = In + O(x) 2 Mn(C[[x]]) satis…es
m 1
where In is the identity matrix of size n.
L =
JM
j=1
jInj
eF (0) = In
+ Jnj where Inj is the identity matrix of size nj,
Jnj =
2
0
6
6.
6
4.
1
. .. . ..
.. .
3
0
7
. 7
15
0 0
is an irreducible Jordan block of size nj (Jnj = 0 if nj = 1) and the eigenvalue j satis…es
Q 1
x =
JM
j=1
qj
1
x
rank of the system)
Re( j) 2 [0; 1[
Inj is a diagonal matrix with polynomial entries in 1
x
qj
1
x
= aj;r
x r
aj;r 1
xr 1
aj;1
x
1
2
x C
h
1
i
x
of degree r (the
or qj 0. It is well known that such conditions can be ful…lled by means of algebraic gauge
transformations (see [BJL79] for example).
In addition, without loss of generality, we assume that
(2) 1 = 0 and q1 0;
conditions that can always be ful…lled by means of a change of variable Y = x 1 e q1(1=x) Z.
Finally, the qj’s are not supposed to be distinct; however, the assumption of a single level
equal to r is equivalent to the conditions
(3)
qj 6 0 () aj;r 6= 0 (the exact degree is r)
qj q` () aj;r = a`;r:
2

When r = 1, M. Loday-Richaud and myself have made explicite in [L-RR] formulæ linking
the resurgence multipliers of the Borel transform b F ( ) to the Stokes multipliers (those entries of
Stokes matrices that are not, a priori, trivial). Such formulæ have been ful…lled from the precise
description of singularities in the Borel plane that we have derived from a proof of the summabilityresurgence
of solutions following Ecalle’s method by perturbation and majorant series.
The aim of this paper is to generalize these results to System (1) with arbitrary rank r 2. We
treat this case by performing a reduction of the rank to one. Doing so we are not given a system
of pure level r = 1 because, in general, there appears other levels smaller than one. Even so, the
formal series solutions of the reduced system keep being 1-summable, the singularities appearing
in the Borel plane are no more singular regular but generically irregular and the analysis gets much
more complicated. Formulæ thus obtained connect the resurgence multipliers of Borel transforms
of solutions of the reduced system and the Stokes multipliers of the initial system (1). Actually,
splitting e F (x) into J column-blocks accordingly the Jordan structure of L, we can restrict
ourselves, without loss of generality, to the correspondence between the resurgence and Stokes
multipliers associated with the …rst column-block of e F (x) that we denote by e f(x). Indeed, any of
the J column-blocks of e F (x) can be positioned at the …rst place by means of a permutation P on
the columns of e Y (x) 1 .
The organization of this paper is as follows. In Section 1, we collect informations on the
Borel transformation and summablity-resurgence needed in the sequel. In Section 2, we apply
rank reduction to System (1). In Section 3, we provide a proof of the summability-resurgence of
formal solutions of the reduced system following Ecalle’s method by perturbation and majorant
series. Section 4 is devoted to the analysis of singularities in the Borel plane. In Section 5, we state
formulæ linking the Stokes multipliers to the resurgence multipliers. These ones are illustrated in
Section 6 with three examples.
1 Preliminaries
1.1 Formal Borel transformation
One calls formal Borel transformation the linear operator
eB : e'(t) = X
'mt m 2 tC[[t]] 7 ! b'( ) = X
m 1
1 The new formal fundamental solution is then
m 1
'm
eY (x)P = e F (x)P x P 1 LP e P 1 Q(1=x)P
3
m 1
(m 1)!
2 C[[ ]]

etween the C-vector spaces tC[[t]] and C[[ ]].
Recall that this operator extends, by setting e B(1) = the Dirac distribution at the origin, into
2 d
an isomorphism from the di¤erential C-algebra C[[t]]; +; ; t to the di¤erential C-algebra
dt
C C[[ ]]; +; ; that changes ordinary product e' e into the convolution product
b' b Z
( ) =
0
b'( ) b ( )d
2 d
(the integral is applied separately to each term of the series) and changes derivation t
dt into
multiplication by . It also changes multiplication by 1
d
into derivation allowing thus to extend
t d
the isomorphism e B from the meromorphic series C[[t]][t 1 ] to C[ (k) ; k 2 N] C[[ ]].
When the series e' is 1-Gevrey 2 (we denote e'(t) 2 C[[t]]1), the formal Borel transform b' is a
convergent series and its sum (also denoted b') is called the minor of e'. In the special case when
e' is convergent (e'(t) 2 Cftg), the minor b' is an entire function that grows at most exponentially
at in…nity.
1.2 Summable-resurgent series
A summable-resurgent series is a 1-Gevrey series e'(t) whose the analytic continuation of the minor
b'( ) outside its disc of convergence lives on a Riemann surface ^
Cn (b') and grows exponentially
at in…nity (cf. De…nition 1.1 below). In the linear single level case we can restrict ourselves to
the case when the set of singularities of the analytic continuation is (b') := f0; !1; :::; !mg where
!1; :::; !m 2 C . For more general de…nitions, we refer to [E85] or [Sau06].
Given > 0, smaller than the minimal distance between elements of (b'), we call -sectorial
region a domain of the universal covering ^
Cn (b') of Cn (b') lying at a distance at least from
(b') and composed of the three following parts:
an open sector of bounded width at in…nity;
a neighborhood of 0 in the principal sheet, say, an open disc centered at 0;
a tubular neighborhood of a piecewise-C 1 path connecting 0 to after a …nite number of
turns around all or part of points of (b').
2 The coe¢ cients 'm satisfy: there exists C; A > 0 such that j'mj CA m m! for all m 0.
4

a -sectorial region
De…nition 1.1 A 1-Gevrey series e'(t) 2 C[[t]]1 is said to be summable-resurgent with singular
support (b') if it satis…es the two following conditions:
i. The minor b'( ) can be analytically continued to any -sectorial region of the above type.
One says then that b' is an endless continuable analytic function with singular support (b').
ii. On all -sectorial region of the above type, b' grows at most exponentially.
The type of exponential growth depends, in general, on the region and is not bounded when
the width of goes to in…nity.
De…nition 1.2
1. The 1-Gevrey series e'(t) 2 C[[t]]1 is said to be resurgent with singular support (b') if it
satis…es Condition (i) of De…nition 1.1.
2. It is said to be summable if it satis…es Conditions (i) and (ii) in restriction to sectors of
C itself (sectors in the usual meaning) issuing from 0 and avoiding points of (b') at some
distance > 0.
3. It is said to be summable in a direction if it satis…es Conditions (i) and (ii) in restriction
to some sector of C bisecting and avoiding points of (b') at some distance > 0.
Note that a summable-resurgent series is therefore both resurgent and summable but the converse
is false.
The set of summable series in a direction is denoted by Cftg1; . Recall that the sum s (e') of
e' 2 Cftg1; is given by
Z 1ei s (e')(t) := b'( )e
=t
d
0
5

2 d
and the operator s is an injective morphism from the di¤erential C-algebra Cftg1; ; +; ; t
dt
to the di¤erential C-algebra of holomorphic functions on sectors with vertex 0, bisecting direction
and larger than a half-plane ([MR89]).
De…nitions 1.1 and 1.2 extend naturally to the set
n o
e 1
(t) = p(t ) + e'(t) ; p(t) 2 C[t] ; e'(t) 2 C[[t]]1 with (b') …nite
of 1-Gevrey meromorphic series with …nite singular support. Indeed, by Borel transformation,
b ( ) = X
k=0
pk (k) + b'( ) ; 2 N ; pk 2 C,
and the singular support of b is ( b ) = (b').
In particular, if e' is summable-resurgent (resp. resurgent, summable) with singular support
(b') and if m is meromorphic at 0, then the product m e' keeps being summable-resurgent (resp.
resurgent, summable) with singular support (b').
1.3 Extension of formal Borel transformation and summation operator
To hold the case of systems with single arbitrary level in full generality, it is necessary to extend
the operators e B and s to formal expansions of the form
e'(t) = X
m'm(t)
m 0
where, for all m 0, m 2 C and 'm belongs to the algebra
(4) < t ; log t ; e P (t 1=r ) ; 2 C and P (t) 2 tC[t] ; deg(P ) r 1 >
We recall in this section some points of the general theory of resurgent functions that will be
used in this paper. For more details, we refer to [E85] or [Sau06].
Denote by
C1 the Riemann surface of the logarithm, i.e., the universal covering of C of base-point 1;
C1f g the set of germs of (multiform) analytic function at 0 2 C1, i.e., the set of germs of
analytic function in a domain of the form
ae i
with a continuous function h : R !]0; +1[.
; 0 < a < h( ) ; 2 R C1
6

The set Cf g of germs of analytic function at 0 2 C identi…es itself to the subset of C1f g formed
by germs of regular analytic function at 0 2 C1. The quotient
C := C1f g=Cf g
is called set of microfunctions at 0 and the canonical mapping from C1f g to C is denoted by
“can”. The variation of 2 C is the unique ' 2 C1f g de…ned by
'( ) := var( )( ) = ( ) ( e 2i )
for any arbitrary representative of in C1f g, i.e., such that can( ) = .
1.3.1 Resurgence constant
Let '(t) 2 C1ftg be with a subexponential growth at 0, i.e., such that
(5) lim jtj log (j'(t)j) = 0 uniformly on all sectors 1 < arg(t) < 2.
jtj!0
The Borel transform
B (')( ) := 1
2i
Z
'(t)e
(the path is given Figure 1) of '(t) in any direction de…nes a germ of analytic function at
0 2 C1 that can be analytically continued on C1 with an exponential growth at in…nity uniformly
on all sectors 0 1 < arg( ) < 0 2. The analytic continuation b' of any B (') is called minor of '.
Note that the formal Borel transformation e B of Section 1.1 is obtained by applying B separately
to each term of the series.
The set of '(t)’s satisfying Condition (5) is called set of resurgence constants 3 and is denoted
by Rc. Note that all element of the algebra (4)is a resurgence constant.
Figure 1
3 The terminology “constant” is justi…ed by the fact that e' having no singularity, except 0, in C1, all its alien
derivatives are zero.
7
=t dt
t 2

Given ' 2 Rc, its minor e' can be non-integrable at 0. For example, '(t) = t 1=2 has for minor
b'( ) = 2 p 3=2 . Consequently, the convolution product and the Laplace transformation L
of Section 1.1 can be no more used in general. To circumvent this di¢ culty, one proceeds as follows:
for any arbitrary u near 0 2 C1, one de…nes
called major of ';
one denotes by
u( ) := 1
2i
the class of majors of ' modulo Cf g;
Z 1=u
0
'(t)e
=t dt
t 2
['] := can( u) 2 C
one extends the Borel transform B (') of any ' 2 Rc into
Note that the minor b' is the variation of ['].
B (') := ([']; b')
2 C1f g
Doing that, the operator B is an isomorphism from the di¤erential C-algebra 2 d
Rc; +; ; t
dt
to the di¤erential C-algebra C1; +; ~; of microfunctions at 0 the variation of which can be
analytically continued on C1 with an exponential growth at in…nity uniformly on all sectors
0
1 < arg( ) < 0 2 d
2. It changes derivation t
product ' into the convolution product
dt
['] ~ [ ] := can( u )
de…ned like the class modulo Cf g of the truncated convolution product
( u )( ) :=
Z u
u
into multiplication by and changes ordinary
( ) ( )d 2 C1f g
(u arbitrarily near 0 2 C1 such that 2]0; u[ and arg( u) = arg ) of any arbitrary
representative and of ['] and [ ] in C1f g respectively.
The reciprocal of B is the Laplace transformation operator L de…ned for all 2 C1 by
Z
(6) L ( )(t) :=
0 ;"
( )e
Z 1ei =t
d +
"e i
b'( )e
where ( ) 2 C1f g is any representative of , b'( ) 2 C1f g is the variation of and 0 ;" is the
circle with center 0 and radius " > 0 covered once in counterclockwise from "e i( 2 ) .
8
=t d

For simplicity, one can compare the class ['] to a speci…c major that one denotes by maj(').
One assumes besides that this major is holomorphic on C1 and grows exponentially at in…nity
uniformly on all sectors 0 1 < arg( ) < 0 2 (such a major always exists, cf. [E85]). Doing that, for
all '; 2 Rc,
(7)
(8)
2 d'
maj t
dt
= maj(')
maj(' ) = maj(') ~ maj( )
and the Laplace transform L ([']) becomes
Z
L (['])(t) = L (maj('))(t) =
maj(')( )e
where is the Hankel loop around 0 like shown in Figure 2. Moreover,
L (maj(')) = ' and L (maj(') ~ maj( )) = '
Figure 2
Complete this section by some classical formulae:
function ' minor b' major maj(')
t m with m 2 N
t with 2 CnZ
m 1
(m 1)!
1
( )
t m with m 2 N 0 (or (m) )
9
1
( )(1 e 2i )
m 1
(m 1)!
=t d
log
2i
= 1
2i ei (1 ) 1
( 1) m m!
2i m+1

More generally, for all p 2 N,
8
maj(t log p ><
t)( ) =
>:
pX
k=0
p+1 X
k=0
k
k
1 log k
1 log k
1.3.2 Extension of formal Borel transformation
Given a formal expansion
(9) e' = X
m 0
if 2 CnN
if 2 N
m'm
where k; k 2 C
where m 2 C and 'm 2 Rc for all m 0, one formally de…nes the minor b' and a major of e'
by setting
b' := X
m b'm and := X
mmaj('m)
Denote by
m 0
m 0
H the set of formal expansions (9) satisfying ( ) 2 C1f g;
H ;exp the set of e' 2 H, the minors b' of which can be analytically continued on unlimited
sectors bisecting direction with an exponential growth of order one at in…nity.
The formal Borel transformation e B of Section 1.1 extends to H by setting
eB(e') = ([e']; b')
where [e'] = can( ) and b' = var([e']). Note that, thus extended, e B can be seen like a linear
operator from the di¤erential C-algebra 2 d
H; +; ; t
dt
to the di¤erential C-algebra C; +; ~; .
2 d
In particular, H ;exp; +; ; t
dt
is a di¤erential C-algebra. Note also that one has the inclusions
Rc H ;exp H
As previously, one compares the class [e'] to a speci…c major that one denotes by maj(e').
In the case when e' 2 H ;exp, one de…nes the Laplace transform L ([e']) by Identity (6); it is
an analytic function on a sector with vertex 0, bisecting direction and opening larger than a
half-plane. The Laplace transform L thus de…ned is compatible with the convolution product ~:
L (maj(e') ~ maj( e )) = L (maj(e'))L (maj( e ))
10

Note that if one assumes that maj(e') can be analytically continued on unlimited sectors bisecting
directions and 2 with an exponential growth of order one at in…nity (one can always choose
it like this, cf. [E85]), one has
Z
L ([e'])(t) = L (maj(e'))(t) = maj(e')( )e
=t
d
Complete this section by a proposition that will be used in the study of singularities and that
describes the major maj(e') in the special case when e' reads e (t)t log p t with e (t) 2 C[[t]]1, 2 C
and p 2 N.
Proposition 1.3 Given e (t) 2 C[[t]]1 a 1-Gevrey series, 2 C and p 2 N,
1. e (t)t log p t 2 H;
2. one can choose for major
8
maj( e (t)t log p ><
t)( ) =
>:
pX
k=0
p+1 X
k=1
qX
k( )
`=0 k=0
k( ) log k
pX
1 log k
`;k +` 1 log k +
p+1 X
k=1
k( ) log k
where k( ) is analytic at 0 and `;k is constant for all ` and k.
if 2 CnZ
if 2 N
if = q 2 N
Moreover, if e is summable-resurgent with singular support ( b ), then the analytic functions k
are endless continuable with singular support ( b ) and grow exponentially on all -sectorial regions
.
1.3.3 Application to the summation
Let
e'(t) = X
m 0
mt m 2 C[[t]]1
be a 1-Gevrey series. The minor b' and the major maj(e') being de…ned by
8
b'( ) =
><
0 +
>:
X
m 1
m
m
(m
1
1)!
2 Cf g
maj(e')( ) = 0
2i +
X m
m
(m
!
1 log
1)! 2i
2 C1f g
m 1
11

it results from Section 1.3.2 the following elementary lemma:
Lemma 1.4
1. e'(t) 2 Cftg1; () e'(t) 2 H ;exp.
2. If e'(t) 2 Cftg1; , then the sum s (e') of e' in the direction
Z
is given by
s (e')(t) = maj(e')( )e
=t
d
We deduce from it the two propositions below that will be used to state the connections between
Stokes and resurgence multipliers:
Proposition 1.5 Let e'(t) 2 Cftg1; be a summable series in the direction and 2 Rc.
1. e' 2 H ;exp.
2. L (maj(e' )) = s (e') .
Proof. The …rst point is straightforward from Lemma 1.4, point 1, and the structure of algebra
of H ;exp. As for the second point, it results from identities
and Lemma 1.4, point 2.
L (maj(e' )) = L (maj(e') ~ maj( )) = L (maj(e'))L (maj( ))
Proposition 1.6 Let 2 C be and e'(t) 2 C[[t]]1 a 1-Gevrey series.
If e'(t)t 2 H ;exp for any direction , then e' is summable in the direction and
L (maj(e'(t)t )) = s (e')t
Proof. Since t 2 Rc H ;exp, the structure of algebra of H ;exp implies e' 2 H ;exp. The series
e' is therefore summable in the direction (Lemma 1.4) and we conclude by Proposition 1.5.
2 Rank reduction
Denote by
A(x) = X
Amx m
m 0
and, for k = 0; :::; r 1, A k (x) = X
Fix, once and for all, a r-th root x = t 1=r of t and denote by
:= exp 2i
r
12
m 0
Ak+mrx m

The r-reduced system of System (1) reads ([L-R01], [Lu72], [Tu63])
2 dY
(10) rt
dt
= A(t)Y
where A(t) is the blocked matrix of size rn rn de…ned by
2
6
A(t) = 6
4
A 0 (t) tA r 1 (t) tA 1 (t)
A 1 (t) A 0 (t)
.
. ..
.. .
. ..
. ..
.
.. . 0 A (t) r tA 1 (t)
Ar 1 (t) A1 (t) A0 (t)
To e Y (x) corresponds the formal fundamental solution
>:
eY (t) = e F (t) e Y 0(t)
of System (10) de…ned by
8 2
eF
6
eF ><
(t) = 6
4
0 (t) t e F r 1 (t) t e F 1 eF
(t)
1 (t)
.
.
Fe 0 (t)
. ..
. ..
. ..
. ..
. ..
e0 F (t)
.
.
tFe r 1 (t)
eF r 1 3
7
5
(t) Fe 1 (t) Fe 0 (t)
and
8
><
>:
2
6
eY
6
0(t) = 6
4
t
Qk (t) := Q
t
eF k (t) = X
m 0
Fk+mrt m ; k = 0; :::; r 1
.
.
3
7
5
Mr
1
ktIn
t L
r e Q0(t) t L
r L e Q1(t) t L
r (r 1)L e Qr 1(t)
L In
r e Q0(t) t
.
L (r 1)In
r e Q0(t) t
k=0
L In
L In
L In Q1(t) (r 1)(L In) Qr r e t r e 1(t)
.
. ..
L (r 1)In
r L In e Q1(t) t
1
k t 1=r ; k = 0; :::; r 1
.
L (r 1)In
r (r 1)(L (r 1)In) e Qr 1(t)
Note that e F (0) is not equal to Irn in general, but is always invertible since it is the lower triangular
blocked matrix
2
In
6
eF
6 F1
(0) = 6
4 .
0
. ..
. ..
. ..
. ..
3
0
7
. 7
0 5
Fr 1 F1 In
13
3
7
5

A normal form of System (10) reads
2 dY
rt
dt
= A0(t)Y
where A0(t) is the blocked matrix of size rn rn de…ned by
2
rar + tL
6
6(r
1)ar 1
6
A0(t) = 6 .
6
4 .
ta1
rar + tL
. ..
tIn
.. .
. ..
.. .
. ..
rar + tL (r 2)tIn
(r 1)tar
.
.
ta1
1
with, for all k = 1; :::; r,
a1 (r 1)ar 1 rar + tL (r 1)tIn
ak =
JM
j=1
aj;kInj
The formal series e F (t) is therefore uniquely determined by the transition system
2 dF
(11) rt
dt
= A(t)F F A0(t)
jointly with the initial condition F (0) = e F (0) ([BJL79]).
In the sequel, we are only interested in the column-block e f formed by the …rst n1 columns of
eF . Recall that the aim of this paper is to make explicit connections between the Stokes multipliers
of the initial System (1) associated with e f and the resurgence multipliers of the formal Borel
transform b f of e f.
3 Summability-resurgence of e f
The main purpose of this section is to prove the following result:
Theorem 3.1 The formal series e f is summable-resurgent with singular support the points aj;r,
j = 1; :::; J (leading terms of polynomials qj’s).
Theorem 3.1 is proved below following Ecalle’s method by regular perturbation and majorant
series quoted in [E85]. Such a proof having already been detailed in the case of systems with single
level r = 1, we only give the main points and we refer to [L-RR] for more details.
14
3
7
5

3.1 Prepared system
In order to simplify the proof of Theorem 3.1, we prepare System (10) by means of a suitably
polynomial gauge transformation. Recall that such a gauge transformation does not a¤ect the
conclusion of Theorem 3.1.
Let
P (x) = In + P1x + ::: + P2r 1x 2r 1 2 Mn(C[x])
be satisfying P (x) e F (x) = In + O(x 2r ). Note that the gauge transformation Y 7 ! P (x)Y allows
to write the matrix A(x) in the form
A(x) =
rX
k=1
kakx r k + x r L + B(x) where B(x) 2 Mn(x 2r Cfxg)
More precisely, splitting the matrix B(x) = Bj;` (x) into blocks accordingly the Jordan structure
of L, the assumption of single level (Conditions (3)) asserts that
(12) B j;` 8
< O(x
(x) =
:
2r ) if aj;r 6= a`;r
O(x3r ) if aj;r = a`;r
By rank reduction, the corresponding gauge transformation Y 7 ! P (t)Y with
2
6
(13) P (t) := 6
4
allows to assume that
(14)
In + tPr tPr 1 + t 2 P2r 1 tP1 + t 2 Pr+1
P1 + tPr+1
and to write A(t) in the form
.
.
In + tPr
. ..
. ..
. ..
. ..
.. . In + tPr tPr 1 + t 2 P2r 1
Pr 1 + tP2r 1 P1 + tPr+1 In + tPr
e F (t) = Irn + O(t 2 )
(15) A(t) = A0(t) + B(t) where B(t) 2 Mrn(t 2 Cftg)
More precisely, splitting successively the matrix B(t) into r2 blocks B (u;v) (t) of size n n, then
each B (u;v) h
(t) = B (u;v)j;` i
(t) accordingly to the Jordan structure of L, Normalizations (12) imply
(16) B (u;v)j;` 8
O(t
><
(t) =
2 ) if 1 v u r and aj;r 6= a`;r
O(t4 ) if 1 u < v r and aj;r = a`;r
>:
O(t 3 ) if not
15
.
.
3
7
5

Under Assumptions (14) and (15), e f is uniquely determined by the …rst n1 columns of System
(11) jointly with the initial condition e f(0) = Irn;n1 (the …rst n1 columns of the identity matrix Irn
of size rn). Hence, the system
2 df
(17) rt
dt A0(t)f + tfJn1 = Bf
since 1 = 0 and a1;k = 0 for all k 2 f1; :::; rg (Conditions (2)).
On the other hand, we adopt from now on the following notations: given a matrix M of size
rn p with any p 2 N , we split successively M into r row-blocks M u; accordingly the block
structure of A0, then each M u; into J row-blocks M u;j; accordingly the Jordan structure of L.
The notation M u;j;`; refers to the `-th row in M u;j; ; the notation M ;q and M u;j;q to the q-th
column in M and M u;j; respectively. M u;j;`;q denotes the entry of M at row ` in multi-row-blocks
(u; j) and column q.
3.2 Proof of Theorem 3.1
Following Ecalle, we consider, instead of System (17), the regularly perturbed system
2 df
(18) rt
dt A0(t)f + tfJn1 = Bf
ful…lled by substituting B for B. An identi…cation of equal powers of shows that this system
admits a unique formal series solution of the form
ef(t; ) = X
ef m(t) m
m 0
satisfying e f 0(t) = Irn;n1 and e f m(t) 2 Mrn;n1(t2C[[t]]) for all m 1. The proof proceeds by
induction on m and is similar to the one detailed in [L-RR, Section 2]. In particular, for …xed
j 2 f1; :::; Jg and m 1, System (18) shows that the system satis…ed by the e f u;j;`;q
m ’s reads, for
u = 1; :::; r, ` = 1; :::; nj and q = 1; :::; n1,
(19) rt 2 d
dt
aj;r e f u;j;`;q
m t( j u + 1) e f u;j;`;q
m
Xu
1
(r + k u)aj;r+k u e f k;j;`;q
m
k=1
16
t
rX
k=u+1
(k u)aj;k u e f k;j;`;q
= te f u;j;`+1;q
m te f
m
u;j;`;q 1
m
+ B u;j;`; e f ;q
m 1

It turns out then, due to Normalizations (16), that for all u, j and m 1,
ef u;j;
2m 1(t) = O(t 3m 1 ) and
8
< O(t
u;j;
fe 2m (t) =
:
3m )
O(t
if aj;r = 0
3m+1 ) if aj;r 6= 0
Thus the series e f(t; ) can be rewritten as a series in t with polynomial coe¢ cients in . Consequently,
by unicity, the formal series e f(t) corresponds to e f(t; 1). Moreover, for all and in
particular for = 1, the series e f(t; ) admits a formal Borel transform with respect to t de…ned
by
'( ; ) = Irn;n1 + X
'm( ) m
where 'm( ) is the formal Borel transform of e f m(t) for all m 1. System (19) implies a similar
system on the entries of 'm: for …xed j 2 f1; :::; Jg and m 1, the 'u;j;`;q u = 1; :::; r, ` = 1; :::; nj and q = 1; :::; n1,
m ’s satisfy for all
(20) r( aj;r) d'u;j;`;q m
d
X
with ' 0 = Irn;n1. More precisely,
( j u + 1 r)' u;j;`;q
m
u 1
d'
(r + k u)aj;r+k u
k=1
k;j;`;q
m
d
m 1
rX
(k u)aj;k u'
k=u+1
k;j;`;q
m
u;j;`;q 1
m 'm = ' u;j;`+1;q
+ d b B u;j;`;
d
' ;q
m 1
1. when aj;r = 0, the normalization of e Y (x) and the assumption of single level (Conditions (2)
et (3)) imply aj;k = 0 for all k 2 f1; :::; r 1g and System (20) reads
(21) r d'u;j;`;q m
d
( j u + 1 r)' u;j;`;q
m
u;j;`;q 1
m 'm = ' u;j;`+1;q
2. when aj;r 6= 0, System (20) is much more involved since the polynomial
qj
1
x
:= aj;r 1
x r 1 :::
aj;1
x
is no more necessarily equal to 0. Actually, System (20) reads
dZm;j;q
(22) Rj
d
where we denote by:
= SjZm;j;q + Tm;j;q Zm;j;q 1
17
+ d b B u;j;`;
d
' ;q
m 1

(a) Zm;j;q the rnj-dimensional column-vector de…ned by
2 3
Zm;j;q :=
6
4
'1;j;q m
.
' r;j;q
m
7
5 and Zm;j;0 := 0
(b) Tm;j;q the rnj-dimensional column-vector de…ned by
2
d
6
Tm;j;q := 6
4
b B 1;j;;
d
' ;q
.
d
m 1
b B r;j;;
3
7
5
and
2
d
6
T1;j;q := 6
4
b B 1;j;;q
d.
d b B r;j;;q
3
7
5
d
' ;q
m 1
(c) Rj and Sj the rnj-dimensional square matrices de…ned by
2
6
Rj := 6
4
r(
(r
aj;r)Inj
1)aj;r 1Inj
.
.
r(
0Inj
aj;r)Inj
.. .
. ..
.. .
. .. r(
.. .
aj;r)Inj
0Inj
.
.
0Inj
and
Sj :=
2
6
4
aj;1Inj (r 1)aj;r 1Inj r( aj;r)Inj
Lj rInj aj;1Inj (r 1)aj;r 1Inj
0Inj Lj (r + 1)Inj
.
.
..
.
. ..
. ..
. ..
.. . Lj (2r 2)Inj aj;1Inj
0Inj 0Inj Lj (2r 1)Inj
Note that Rj and Sj are independent of m and q.
Fix now > 0 and a -sectorial region associated with ( b f) = faj;r ; j = 1; :::; Jg as above
(Section 1.2). We must prove that
the series b f( ) = '( ; 1) = Irn;n1 + X
'm( ) is convergent at 0 and can be analytically
continued to ;
m 1
bf( ) grows at most exponentially on at in…nity.
These two properties are ful…lled by applying a technique of majorant series similar to the one
detailed in [L-RR, Section 2]. Consequently, we only provide below a sketch of their proof.
Due to the de…nition of the two following properties hold:
18
d
.
.
3
7
5
3
7
5

(25)
1. There exists a constant a > 0 such that for all 2
(23) jarg( )j a
2. There exists a constant K > 0 such that for all 2 , there is a piecewise-C 1 -path
parametrized by arc length from 0 to such that the arc length s of all 2 satis…es
(24) j j s K j j
(j j denotes the modulus of the projection of on C).
Let us, for u = 1; :::; r and j = 1; :::; J, the perturbed system
8
1
CK
1 Re
j
r
u 1
r
gu;j; = Jnjgu;j; + gu;j; Jn1
><
>:
j
r
= t
u 1
r
rX
k=u+1
1 t g u;j;
+ 1
CK Iu;j;
rn;n1
Xu
1
jaj;r+k uj gk;j; k=1
2I u;j;
rn;n1 Jn1 + jBju;j;
t
g if aj;r = 0
jaj;k uj g k;j; + Jnj tgu;j; + tg u;j; Jn1 + jBj u;j; g if aj;r 6= 0
where the unknown g is, like b f, a rn n1-dimensional matrix splitted in multi-row-blocks g u;j; ,
C is the positive constant
C := max
1 j J exp (2a jIm( j)j)
and jBj denotes the series B in which the coe¢ cients are replaced by their module.
System (25), like System (18), admits a unique formal series solution
eg(t; ) = X
eg m(t) m
m 0
such that eg 0(t) = Irn;n1 and eg m(t) 2 Mrn;n1(t 2 C[[t]]) for all m 1. Moreover, this series satis…es
the three following properties:
for all u = 1; :::; r, j = 1; :::; J and m 1,
eg u;j;
2m 1(t) = O(t 3m 1 ) and eg u;j;
8
<
2m (t) =
:
19
O(t 3m ) if aj;r = 0
O(t 3m+1 ) if aj;r 6= 0;

for all m 1, eg m(t) has non-negative coe¢ cients;
for all , eg(t; ) determines an analytic function in t at 0.
Hence, the formal series eg(t) = eg(t; 1) is convergent at 0 as a series in eg m(t) as well as a series in
t. Consequently, its formal Borel transform
( ) = Irn;n1 + X
( m( ) 2 Mrn;n1( R + [[ ]]) is the formal Borel transform of eg m(t) for all m 1) is an entire
function with exponential growth at in…nity.
Theorem 3.1 results then from Lemma 3.2 below.
m 1
m( )
Lemma 3.2 The series (K ) is a majorant series for b f( ) = Irn;n1 + X
'm( ).
Proof. Prove the following inequalities: for all m 1, 2 , u = 1; :::; r, j = 1; :::; J and
q = 1; :::; n1,
(26) ' u;j;q
m ( )
(27)
><
>:
u;j;q
m (s )
u;j;q
m (K j j)
The second inequality is straigthforward from Condition (24) and the fact that has nonnegative
coe¢ cients. As for the …rst inequality, it results from System (27) below satis…ed by the
entries
(u;j;`);q
m :
8
1
CK
1 Re
j
r
u 1
r
u;j;`;q
m = u;j;`+1;q
m
d u;j;`;q
m
d
j
r
Xu
1
k=1
u 1
r
jaj;r+k uj
1
u;j;`;q
m
d k;j;`;q
m
d
=
+ u;j;`;q 1
+ u;j;`+1;q
m +
m
rX
k=u+1
20
jaj;k uj
u;j;`;q 1
m
+ dd jBj u;j;`;
d
k;j;`;q
m
+ dd jBj u;j;`;
d
m 1
u;j;q
m
;q
m 1 if aj;r = 0
;q
m 1 if aj;r 6= 0

In the case when aj;r = 0, the solution of Equation (21) given
(28) ' u;j;`;q
m ( ) =
1
r
j
r
u 1
r
Z
1
0
d b B u;j;`;
d
' ;q
m 1( ) + ' u;j;`+1;q
m ( ) '
u;j;`;q 1
m
( )
!
j u 1
+ r r d ;
Conditions (23) and (24) allow, by majorizing (28), to prove the …rst inequality of (26).
To conclude in the case when aj;r 6= 0, we rewrite System (27) in a similar form to System (22)
and we apply Grönwall Lemma to ' u;j;`;q
m ( (s)). This achieves the proof of Lemma 3.2.
4 Singularities of the Borel transform b f
The aim of this section is to describe the precise possible form of the singularities aj;r 6= 0. Contrary
to the systems with single level 1 (see [L-RR, Section 3]), these singularities are no more singular
regular but generically irregular. However, the kind of “irregularity” of each singularity can be
directly read on the initial formal fundamental solution e Y (x) since it depends only on polynomials
qj.
Following Ramis, we call front of ! the set
F r(!) := fqj ; aj;r = !g
Under our assumption of single level (Conditions (3)), the front of each singularity ! of b f is a
singleton:
! 1
1
!;r 1
F r(!) = + q! with q! :=
xr x
x xr 1 !;1
:::
x
In the case when q! 0 (we refer to such a singularity ! as a singularity with monomial front),
the singularity ! is regular. More precisely, we …nd a simple-rami…ed structure (cf. De…nition
4.1 below) similar to the one already met in the case of systems with single level 1 (see Section
4.1, Theorem 4.2). On the other hand, when q! 6 0, the singularity ! becomes irregular and the
polynomial q! determines completely this irregularity (see Section 4.2, Theorem 4.14).
In order to simplify the notations, we use, from now on, the translation operator ! in !:
! b f( ) = b f( + !)
21

4.1 Singularities with monomial front
Let ! be such a singularity. Theorem 4.2 below shows that ! admits a simple-rami…ed structure
similar to the one already met in [L-RR]. Note that this result is not surprising since, the monomial
!=x r in the initial system (1) providing by rank reduction the monomial !=t, System (10) is
“morally”a system with single level 1 for !.
Before to formulate Theorem 4.2, recall the de…nition of the simple rami…ed singularities.
De…nition 4.1 (simple-rami…ed singularity)
A germ ' is said to be simple-rami…ed at ! 2 C if, modulo germs of analytic function at 0, !'
reads
where
!'( ) =
!;0 X
p=0
!;p log p +
!;0+1
X
h!;p( ) log p + X !; X
(H!; ;p( ) log p )
p=1
2 ! p=0
i. the resurgence coe¢ cients !;p, on the one hand, and h!;p( ) and H!; ;p( ), on the other
hand, are constant and analytic at 0 respectively;
ii. the set of rami…cation exponents ! is a …nite subset of CnZ in which values are distinct
modulo Z;
iii. the logarithmic orders ( !;0; ( !; ) 2 !) are non-negative integers.
Theorem 4.2 (description of singularities with monomial front)
Let ! be any of aj;r 6= 0 with monomial front and q 2 f1; :::; n1g. The q-th column b f ;q
simple-rami…ed singularity at !. Modulo germs of analytic function at 0, ! b f ;q
reads
! b f ;q
N!;0+q
X
( ) =
p=0
;q
N!;0+q+1
X
!;p p
log +
p=1
h ;q
!;p( ) log p + X
2 !
N!; +q
X
p=0
H ;q
!; ;p ( ) logp 1
1
of b f has a
where ;q
!;p 2 Mrn;1(C) and h ;q
!;p( ); H ;q
!; ;p ( ) 2 Mrn;1(Cf g). The set of rami…cation exponents !
22

and the logarithmic orders (N!;0 + q; (N!; + q) 2 !) are given by
8
! =
j
r
u 1
; j 2 f1; :::; Jg and u 2 f1; :::; rg such that
r
><
>:
N!;0 = max nj 2 ; j 2 f1; :::; Jg such that qj
N!; = max nj 2 ; j 2 f1; :::; Jg such that qj
qj
1
x
1
x
1
x
= !
x r and ( j 6= 0 or u 6= 1)
= !
x r and j = 0
!
= and
xr there exists u 2 f1; :::; rg such that
j
r
u 1
r =
Moreover, the resurgence coe¢ cients of b f ;q
at ! are germs of endless continuable analytic function
with singular support the points aj;r ! that grow exponentially at in…nity.
Note that the resurgence coe¢ cients of b f ;q
at ! depend on the path chosen for the analytic
continuation of b f.
Proof. To prove Theorem 4.2, we follow Ecalle’s method by regular perturbation quoted in [E85]
(see also [CNP93] and [Sau06]). Resuming the notations of Section 3, the formal series e f reads
X
(29)
f(t) e 1
= P (t) ef m(t)
where the matrix P is de…ned by (13) and the e f m’s by (19). Thus, after a formal Borel transformation,
Identity (29) becomes
X
(30) f( b 1
) = P (0) + E 'm( )
where E is an entire function (since P 1 (t) is analytic at 0) and the 'm’s are de…ned by (20).
Ecalle’s method consists in showing that, for …xed q 2 f1; :::; n1g, the ' ;q
m ’s have the same structure
at the singularity !, then this structure is transmitted to b f ;q
. The arguments used in this proof
are similar to those detailed in [L-RR, Section 3] for systems with single level 1. Consequently, we
only give the main points.
In order to study the behavior of the ' m’s near !, we need to explicitly know their entries
23
m 0
m 0

with u = 1; :::; r, j = 1; :::; J, ` = 1; :::; nj and q = 1; :::; n1.
Recall that, in the case when aj;r = 0, the 'u;j;`;q m ’s are given by
' u;j;`;q
m
(31) ' u;j;`;q
m ( ) =
1
r
j
r
u 1
r
Z
1
0
d b B u;j;`;
d
' ;q
m 1( ) + ' u;j;`+1;q
m ( ) '
u;j;`;q 1
m
In the case when aj;r 6= 0, the 'u;j;`;q m ’s are determined by System (22). More precisely,
( )
!
j u 1
+ r r d
when aj;r = !, the polynomial qj 2 F r(!) is the monomial !=x r (! is a singularity with
monomial front). Consequently, aj;k = 0 for all k 2 f1; :::; r 1g and System (22) splits into
the di¤erential equations
r( aj;r) d'u;j;`;q m
d
( j u + 1 r)' u;j;`;q
m
u;j;`;q 1
m 'm = ' u;j;`+1;q
+ d b B u;j;`;
d
' ;q
m 1
Therefore, since ' m(0) = 0, it results that, for all u = 1; :::; r, ` = 1; :::; nj and q = 1; :::; n1,
(32) ' u;j;`;q
m
( ) = 1
j
( !) r
r
Z
0
d b B u;j;`;
d
u 1
r
1
' ;q
m 1( ) + ' u;j;`+1;q
m ( ) '
u;j;`;q 1
m
( )
!
( !)
j u 1
+ r r d
when aj;r 6= !, the calculation of the 'u;j;`;q m ’s is much more involved since System (22) has
generally an irregular singularity at aj;r. Denote then by Fj( ) a fundamental solution at 0
of the homogeneous system of (22). Note that Fj( ) is independant of q and m, analytic at 0
(since aj;r 6= 0) and can be analytically continued on the universal covering of Cnfaj;rg. Thus,
solving System (22) by the method of variation of constants, it results that the column-vector
2 3
Zm;j;q =
6
4
'1;j;q m
.
' r;j;q
m
reads, for all u = 1; :::; r and q = 1; :::; n1,
Z
(33) Zm;j;q( ) = Fj( )
0
7
5 ; Zm;j;q(0) = 0
F 1
j ( )R 1
j ( ) (Tm 1;j;q( ) zm;j;q 1( )) d
Note that Fj( ) and R 1
j ( ) are analytic at ! since aj;r 6= !.
24

We are left to prove that, for …xed q 2 f1; :::; n1g,
i. the ' ;q
m ’s, m 1, have the same structure at !: simple-rami…ed with the same set of
rami…cation exponents ! and the same logarithmic orders (N!;0 + q; (N!; + q) 2 !);
ii. this structure is successively transmitted to the series P ;q
m 1 ' m ( ) and b f ;q
.
The …rst point is proved by recursion on m 1 applying Proposition 4.3 below to Identities (31),
(32) and (33). For the second point, we use the uniform convergence of Series P
m 1 ' m( ) stated
in Section 3.2, then we apply Proposition 4.3 again to Identity (30).
As for the properties of the resurgence coe¢ cients, they result from the summability-resurgence of
ef. For more details, we refer to [L-RR, Section 3].
Proposition 4.3 (stability of simple-rami…ed singularities)
Let e be an entire function with valuation v at the origin.
Let ' be a germ of endless continuable analytic function with a …nite singular support .
Assume that ' has a simple-rami…ed singularity at ! 2 with rami…cation exponents ! and
logarithmic orders ( !;0; ( !; ) 2 !).
Then, the convolution product e ' admits at ! a singularity with the same type: simple-rami…ed,
same rami…cation exponents ! and same logarithmic orders ( !;0; ( !; ) 2 !): modulo germs of
analytic function at 0,
!(e ')( ) =
!;0+1
X
p=1
k!;p( ) log p + X !; X
(K!; ;p( ) log p )
2 ! p=0
Note that no terms logp
appears. Moreover, the valuation at 0 of the resurgence coe¢ cient k!;p
is equal to v and the valuation of the resurgence coe¢ cient K!; ;p is v + 1.
When the matrix L is diagonal, the structure of ! is more simpler, but, contrary to systems
with single level 1, there always exists non-integer powers of , included when the matrix L is
trivial:
Corollary 4.4 Assume that the matrix L is diagonal 4 : L = diag(0; 2; :::; n).
Let ! be any of aj;r 6= 0 with monomial front. Then, modulo germs of analytic function at the
origin, ! b f reads
! b f( ) = ! + h!( ) log + X
1
H!; ( )
4 To simplify the notations in this case, we do not write the index q of columns.
25
2 !
1

where ! 2 Mrn;1(C) and h!( ); H!; ( ) 2 Mrn;1(Cf g). The set of rami…cation exponents ! is
given by
! =
j
r
u 1
r
; j 2 f1; :::; ng and u 2 f1; :::; rg such that
4.2 Singularities with non-monomial front
qj
1
x
= !
x r and ( j 6= 0 or u 6= 1)
Fix now a singularity ! of b f with non-monomial front. The previous method does not provide a
description of ! with enough accuracy. Indeed, System (22) has now an irregular singularity at !
and the calculations become more involved. Actually, we can only assert that the singularity ! is
no more singular regular but irregular.
Before to start the study of this singularity, look at a simple example.
4.2.1 Example
Consider the system
3 dY
(34) x
dx
= 0 0
x 4 2 + x Y
with single level 2 and its formal fundamental solution e Y (x) = e F (x)e Q(1=x) de…ned by
eF (x) = I2 + O(x 4 ) and Q 1
x
= diag 0;
The …rst column b f of the formal Borel transform of e F has only one singularity: ( b f) = f1g, and
it is with non-monomial front:
F r(1) =
1
x2 1
x
The behavior of b f at 1 is provided by explicitly calculating the analytic continuation of b f near 1.
The 2-reduced system of System (34) is given by
2 dY
2t
dt =
2
0
6
6t
4
0 0 0
2 0
2
0
0
t
t
0
0 1 t2 3
7
5
2 t
Y
The formal series e f is therefore uniquely determined by the system
2 df
2t
dt =
2
0
6
6t
4
0 0 0
2 0
2
0
0
t
t
0
0 1 t2 3
7
5
2 t
f
26
1
x 2
1
x

jointly the initial condition e 2 3
1
6
f(0) = I4;1 = 60
7
405
0
(cf. System (17)). Consequently, e f reads
2
1
6
ef(t)
6ef
= 6
4
1;2
(t)
0
ef 2;2
3
7
5
(t)
(since L = 0, we omit the index q of columns) where the formal series e f 1;2
and e f 2;2
ef
satisfy
1;2
(t) = e f 2;2
(t) = O(t2 ) and the di¤erential system
8
><
>:
2t 2 de f 1;2
dt
2t 2 de f 2;2
dt
2 e f 1;2
(2 t) e f 2;2
te f 2;2
= t2 ef 1;2
By formal Borel transformation, it results that b f reads
2 3
1
6
bf( ) = 60
7
405
0
+
2
0
6
6bf
6
4
1;2
( )
0
bf 2;2
3
7
5
( )
where b f 1;2
and b f 2;2
satisfy b f 1;2
( ) = b f 2;2
( ) = O( ) and
8
>< 2( 1) b f 1;2
1 b f 2;2
=
>:
2( 1) b f 2;2
+ 1 b f 2;2
In particular, b f 1;2
is the unique analytic solution at 0 of the di¤erential equation
(35)
8
><
2 dy
4( 1) + (6
d
>:
y(0) = 0
7) y = 3 2
Choose a determination of log such that log is a real number when > 0 and solve (35): for
all 2 [0; 1[,
bf 1;2
( ) = (1 ) 3 1
2 e 4(1
Z
)
3
p
1
2
e
1
4(1 ) d
27
0
bf 1;2
= 0
= 0

where the path (0; ) is the segment [0; ]. More precisely, in terms of special functions, b f 1;2
reads
bf 1;2
( ) = 3
0
@
2
2
1
where
+ 1F1
1 3 ; 2 2 ;
1
4(1 )
(1 ) 2 e
1F1
1
2
1
4(1 )
1
A
+ e 1
4 + 3
2 1F1
+1X
3
; ; =
2
n=0
n
n!(2n + 1)
is the con‡uent hypergeometric function with parameters 1
2
1F1
1 3
;
2 2 ;
and 3
2 .
1
4
(1 ) 3
2 e
Therefore, b f admits an irregular singularity at 1. Concerning the irregular parts e
1 3 ; 2 2 ;
1
4(1 ) + 2
1
4(1 ) and
1
4(1 ) , we will show in Section 4.2.3, Proposition 4.15, that they result from the
rami…ed exponentials e
t 1=2
of e Y ful…lled from the exponential e 1=x during the rank reduction.
With this simple example, we see clearly that the structure of singularities with non-monomial
front is generically more involved since irregular parts can occur. In particular, a study of System
(22) similar to the one of Section 4.1 can not provide such a structure with enough accuracy. To
circumvent this di¢ culty, we use an alternative approach based on microfunctions (or majors).
4.2.2 Equations of exponential type and microsolutions
Recall that the set C of microfunctions at 0 is the quotient
C := C1f g=Cf g
The canonical mapping from C1f g in C is denoted by “can”and “var”is the variation. For all
2 C, we denote by C the translated of C in .
Denote by @t := d
dt the derivative with respect to t. Let p = an(t)@ n t + ::: + a1(t)@t + a0(t) be an
equation of exponential type (the coe¢ cients aj are polynomials and the levels of p are not greater
than 1). Its Borel transform bp = bm( )@ m + ::: + b1( )@ + b0( ) is also an equation of exponential
type and the degree of bm( ) is equal to n (the order of p).
Let 1; :::; s be the singular points of bp, i.e., the zeros of bm, and m1; :::; ms their multiplicity
order. The space of microsolutions of bp in j, i.e., the space of 2 C j satisfying bp = 0 in C j , is
mj-dimensional ([Ka83]).
Let be now a direction such that the half-lines d j; issuying from j in the direction are
distinct. If is a microsolution of bp in j, then var( ) = b' is a complete solution of bpby = 0
28

and can be analytically continued with an exponential growth of order one at in…nity along d j; .
Thus, the Laplace transform L j; ( ) (= the translated in j of (6)) of is well de…ned and it is
a solution of py = 0.
Consequently, the solutions of py = 0 can be obtained in this way:
Theorem 4.5 ([E85], [Mal91]) For …xed j 2 f1; :::; sg, denote by ( j;k)1 k mj a basis of microsolutions
of bp in j.
Then, the set L j; ( j;k)’s, j = 1; :::; s and k = 1; :::; mj, is a basis of solutions of py = 0.
The Laplace transformation de…nes an isomorphism between the set of microsolutions of bp and
the set of solutions of p. We use partially this result as follows:
Corollary 4.6 Let (e'1; :::; e'n) be a basis of formal solutions of p:
e'j = e je j=t with j 2 C and e j 2 t j e P(t 1=k j )C[[t]][log t]
with j 2 C and Pj(t) belonging to tC[t] with degree not greater than kj 1.
For all j 2 f1; :::; ng, choose a major maj( e j) 5 of e j and de…ne b j by
b j := j maj( e j)
1. b j is a major of e'j and it de…nes a multiform analytic function in j.
2. A basis of microsolutions of bp is given by (can( b 1); :::; can( b n)).
4.2.3 Description of the singularity !
In order to apply Corollary 4.6, we reduce System (10) into an equation of exponential type using
Cyclic Vector Theorem ([D70]) and Birkho¤ algebraization Theorem ([Si90]).
Let Y = MZ be, M(t) 2 GLrn(Cftg[t 1 ]), a meromorphic gauge transformation that transforms
System (10) into the companion system of a linear di¤erential equation (E) of order rn with
polynomial coe¢ cients. A formal fundamental solution of (E) is
eZ(t) = e G(t) e Y 0(t) where e G(t) = M 1 (t) e F (t)
Thus, the levels of (E) are not greater than 1 and, consequently, (E) is an equation of exponential
type. Note besides that, M(t) being a meromorphic at 0, e G keeps being summable-resurgent and
the singularities of the q-th column of its minor b G are those of the q-th column of b F .
5 maj( e j) is a function of …nite determination.
29

Denote now by e G1;1 the …rst n1 columns of e G. The study of the analytic continuation of b G1;1
near ! allows us to describe the structure of the singularity ! as singularity of b f. Indeed, the
identity e f = M e G1;1 implies on the Borel transforms the identity
(36) b f = c M b G1;1
and c M reads
cM =
vX
k=0
mk (k) + be
where mk 2 Mrn(C) for all k = 0; :::; v and be is an entire function.
To study the analytic continuation of b G1;1 near !, we proceed as follows:
i. we derive from the formal fundamental solution e Z(t) a basis of formal solutions of (E);
ii. applying Corollary 4.6 to this formal basis, we build a basis of microsolutions at ! of the
Borel transform equation ( b E) of (E);
iii. we interpret the analytic continuation of b G1;1 near ! from these microsolutions.
>:
i. A basis a formal solutions of (E)
Recall that e Y 0(t) is the blocked matrix of size rn rn de…ned by
8 2
6
eY
6
><
0(t) = 6
4
t L
r eQ0(t) t L
r LeQ1(t) t L
L In
t r e
(r 1)L Qr r e 1(t)
Q0(t)
.
L In
t r L In Q1(t) e
.
. ..
L In
(r 1)(L In) Qr t r e 1(t)
.
L
t
(r 1)In
r eQ0(t) 3
7
5
L (r 1)In
t r L In Q1(t) e
L (r 1)In
t r (r 1)(L (r 1)In) Qr e 1(t)
Qk (t) := Q
where = e 2i =r .
Split successively e h
G = eG1
structure of e Y 0, then each e Gu =
1
k t 1=r ; k = 0; :::; r 1
h eGu;1
i
eGr into r column-blocks of size rn n accordingly the block
i
into J column-blocks accordingly the Jordan
eGu;J
structure of L. Denote also by eg u;j;q the …rst entry of the q-th column of e Gu;j. Note that, for
…xed j 2 f1; :::; Jg, the eg u;j;q’s, u = 1; :::; r and q = 1; :::; nj, are summable-resurgent with singular
support the points ak;r aj;r, k = 1; :::; J.
A basis of formal solutions of (E) is de…ned by
(ez1; :::; ezr)
30

where, for u = 1; :::; r,
with, for all j 2 f1; :::; Jg and q 2 f1; :::; njg,
8
qX
ezu;j;q(t) =
><
`=1
>:
and
ezu = (ezu;1;1; :::; ezu;1;n1; ezu;2;1; :::; ezu;J;1; :::; ezu;J;nJ )
ehu;j;`(t)t j=r (u 1) j logq ` ( u 1t1=r )
e
(q `)!
qj( u 1t 1=r )
ehu;j;`(t) = eg 1;j;`(t) + ( u 1 t 1=r ) 1 eg 2;j;`(t) + ::: + ( u 1 t 1=r ) (r 1) eg r;j;`(t)
ii. A basis of microsolutions of ( b E)
The singularities of ( b E) are located at the points = aj;r, j = 1; :::; J. Since (E) is an equation
of exponential type, Corollary 4.6 applies: with each formal solution ezu;j;q is associated a
microsolution of ( b E) at aj;r. It is de…ned by can( b u;j;q) where
b u;j;q = maj(ezu;j;q) = aj;r maj
`=1
qX
`=1
ehu;j;`(t)t j=r (u 1) j logq ` ( u 1t1=r )
e
(q `)!
qj( u 1t 1=r !
)
Lemma 4.7 For all u 2 f1; :::; rg, j 2 f1; :::; Jg and q 2 f1; :::; njg,
b
u;j;q = aj;r maj
qX
ehu;j;`(t)t j=r (u 1) j logq ` ( u 1t1=r !
)
~ maj
(q `)!
e qj( u 1t 1=r )
!
where
i. the major
Nu;j;q := maj
qX
`=1
ehu;j;`(t)t j=r (u 1) j logq ` ( u 1t1=r !
)
(q `)!
satis…es the two following properties called Properties (Paj;r ):
(a) Nu;j;q reads
Nu;j;q( ) = X
where 2 C, p 2 N and ' ;p( ) 2 Cf g;
…nite
' ;p( ) log p
(b) the ' ;p’s are endless continuable with singular support the points ak;r aj;r, k = 1; :::; J,
and grow exponentially at in…nity;
31

ii. the major maj e qj( u 1 t 1=r ) is a major of e qj( u 1 t 1=r ) that is holomorphic on C1 and
grows exponentially at in…nity.
Proof. Point (i) is straightforward from Proposition 1.3. Indeed, for all ` 2 f1; :::; qg, e hu;j;` reads
ehu;j;`(t) = eg 1;j;`(t) + ( u 1 t 1=r ) 1 eg 2;j;`(t) + ::: + ( u 1 t 1=r ) (r 1) eg r;j;`(t)
and the eg v;j;`’s, v = 1; :::; r, are summable-resurgent with singular support the points ak;r aj;r,
k = 1; :::; J.
As for Point (ii), it results from Section 1.3.1 and the fact that the exponential e qj( u 1t 1=r ) is a
resurgence constant (the degree in t of qj
u 1 1=r t r 1
is equal to r < 1 at the most).
d
Note that Properties (Paj;r ) are stable by derivation
d
log and , 2 C.
A basis of microsolutions at any singularity of ( b E) is given by
can( b u;j;q) ; 1 u r and 1 q nj
and convolution by polynomials in
j;aj;r=
Consequently, Lemma 4.7 above allows to describe the structure of all microsolutions of ( b E) at :
Lemma 4.8 Let be a microsolution of ( b E) at .
Choose a representative b of in e O , i.e., such that can( b ) = .
Then, the function b translated by reads, modulo germs of analytic function at 0,
b =
Xr
1
k=0
k ~ maj e q!( k t 1=r )
where the k’s satisfy Properties (P ) for all k = 0; :::; r 1.
iii. Analytic continuation of b G1;1 near !
The study of microsolutions of ( b E) at ! allows us to describe the behavior of b G1;1 near !:
Proposition 4.9 Modulo germs of analytic function at 0, ! b G1;1 reads
(37) ! b G1;1 =
Xr
1
k=0
k ~ maj e q!( k t 1=r )
where the k’s satisfy Properties (P!) for all k = 0; :::r 1.
32

To prove this result, we proceed by recursion on columns b G1;1;q, q = 1; :::; n1, of b G1;1 starting
with q = 1. The calculations being similar for all columns, we only detail the case q = 1.
The …rst column e G1;1;1 of e G1;1 reads
2
eg 1;1;1
3
6
eG1;1;1 = 6 .
4
drneg 1;1;1
dtrn 7
5
since the …rst column of the formal fundamental solution e Z(t) of (E) is
2 3
with
6
4
ez1;1;1
.
d rn ez1;1;1
dt rn
ez1;1;1(t) = e h1;1;1(t) = eg 1;1;1(t) + t 1=r eg 2;1;1(t) + ::: + t (r 1)=r eg r;1;1(t)
Identity (37) in restriction to e G1;1;1 is proved in the two following lemmas.
Lemma 4.10 Modulo germs of analytic function at 0, the …rst entry !bg 1;1;1 of ! b G1;1;1 reads
(38) !bg 1;1;1 =
Xr
1
k=0
7
5
k;1 ~ maj e q!( k t 1=r )
where the k;1’s satisfy Properties (P!) for all k = 0; :::r 1.
Proof. Recall that, for all u 2 f1; :::; rg, the major maj(ezu;1;1) de…nes a microsolution of ( b E) at
0. Its variation
bhu;1;1 = bg u;1;1 + u 1 \
t 1=r eg2;1;1 + ::: + (u 1)(r 1) \
t (r 1)=r egr;1;1
is therefore a complete solution of ( b E) and the analytic continuation of b hu;1;1 near ! de…nes a
microsolution of ( b E) at !. Consequently, Lemma 4.8 implies a system of the following form:
modulo germs of analytic function at 0,
8
><
>:
.
!bg 1;1;1+ !
\
t 1=r eg 2;1;1 + :::+ ! \
t (r 1)=r eg r;1;1 =
Xr
1
k=0
!bg 1;1;1 + ! \
t 1=r eg 2;1;1 + ::: + r 1 ! \
t (r 1)=r eg r;1;1 =
!bg 1;1;1 + r 1 ! \
t 1=r eg 2;1;1 + ::: +
(r 1)2
!
33
k;1 ~ maj e q!( k t 1=r )
Xr
1
k=0
\
t (r 1)=r eg r;1;1 =
k;2 ~ maj e q!( k t 1=r )
Xr
1
k=0
k;r 1 ~ maj e q!( k t 1=r )

where the k;u’s satisfy Properties (P!) for all k = 0; :::r 1 and u = 1; :::; r. The solution of this
system achieves the proof.
Lemma 4.11 Denote by b G `
1;1;1 the `-th entry of b G1;1;1, ` = 2; :::; rn.
Identity (38) keeps holding for all `: modulo germs of analytic function at 0,
(39) ! b G `
1;1;1 =
Xr
1
k=0
k;` ~ maj e q!( k t 1=r )
where the k;`’s satisfy Properties (P!) for all k = 0; :::; r 1.
Proof. Recall that, for ` = 2; :::; rn,
(40)
b G `
1;1;1 = e B d` 1 eg 1;1;1
dt ` 1 = d`
d `
Prove Identity (39) for ` = 2. Identity (40) implies that
` 1 d` 2bg 1;1;1
d ` 2
! b G 2 d
1;1;1 = !
2bg 1;1;1
d 2 + ! d
!
2bg 1;1;1
d 2 + 2 dbg 1;1;1
!
d
Thus, applying Lemma 4.10 and the properties of derivative of the convolution product ~, ! b G 2
1;1;1
reads, modulo germs of analytic function at 0,
(41) ! b G 2
1;1;1 =
Xr
1
k=0
d 2
k;1
d 2 ~ maj eq!( k t 1=r ) + !
Xr
1
k=0
d 2
k;1
d 2 ~ maj eq!( kt 1=r )
Xr
1
+ 2
k=0
d k;1
d ~ maj eq!( k t 1=r )
where the d k;1
’s and
d
d2 k;1
d 2 ’s satisfy Properties (P!) for all k = 0; :::; r 1. On the other hand,
multiplication by is a ~-derivation. Hence, for k = 0; :::; r 1,
with
d 2
d 2
k;1
d 2 ~ maj eq!( k t 1=r ) =
d 2
k;1
d 2 ~ maj eq!( kt 1=r ) = d2
= d2
= d2
k;1
d 2 ~maj eq!( kt 1=r ) d
+ 2
k;1
d 2 ~ maj eq!( kt 1=r )
d
d
d
k;1
d
~ maj t2
2 dt eq!( kt 1=r ) (cf. (7))
k;1
2 ~ maj
k;1
2 ~ maj
34
k
r t1 1=r q 0 !
k
r t1 1=r q 0 !
1
k t 1=r
e q!( k t 1=r )
1
k t 1=r ~ maj e q!( k t 1=r )
(cf. (8))

Therefore,
d 2
k;1
d 2 ~ maj eq!( kt 1=r ) = k ~ maj e q!( kt 1=r )
where k satis…es Properties (P!). Indeed, the major
maj
k
r t1 1=r q 0 !
is a polynomial in log and with 2 CnN (cf. Section 1.3.1). Identity (41) achieves the proof
for ` = 2. In the case when ` 3, the calculations are similar and left to the reader.
1
k t 1=r
Conclusion. Description of the singularity ! as singularity of b f
Applying an elementary adequate generalization of Proposition 4.3, we deduce from Proposition
4.9 a …rst description of singularities with non-monomial front of b f:
Proposition 4.12 Let ! be any of aj;r 6= 0 with non-monomial front.
Modulo germs of analytic function at 0, ! b f reads
! b f( ) =
Xr
1
k=0
!;k ~ maj e q!( k t 1=r ) ( )
where the !;k’s satisfy Properties (P!) for all k = 0; :::; r 1.
The following elementary result allows us to precise this description (see Theorem 4.14 below).
Lemma 4.13 Denote by
and de…ne the support of q! by
Denote also
Then,
1. the % polynomials
are distinct two by two;
q!
1
x
:=
!;r 1
x r 1 :::
a!;1
x
supp(q!) := fk 2 f1; :::; r 1g ; !;k 6= 0g
:= pgcd(j 2 supp(q!); r) and % := r 2 f2; :::; rg
q!
1
k t 1=r ; k 2 f0; :::; % 1g
35

2. for …xed k 2 f0; :::; % 1g, only the polynomials
are equal to q!
1
k t 1=r .
q!
1
k+`% t 1=r ; ` 2 f0; :::; 1g
Theorem 4.14 Let ! be any of aj;r 6= 0 with non-monomial front.
Modulo germs of analytic function at 0, ! b f reads
! b f( ) =
% 1 X
k=0
'!;k ~ maj e q!( k t 1=r ) ( )
where the '!;k’s satisfy Properties (P!) for all k = 0; :::; % 1.
Theorem 4.14 is su¢ cient to make explicit connections between Stokes and resurgence multipliers.
However, we will improve this result in the next section (see Corollary 5.10) precising the
possible form of the '!;k’s at 0.
The majors maj e q!( k t 1=r ) having an irregular singularity at 0, this one is transmitted to
! b f and we …nd the result on singularities with non-monomial front announced at the beginning of
Section 4. A more precise study of this irregularity requires to know explicitly these majors. Their
e¤ective calculation can be performed by means of Theorem 4.5. Indeed, it is easy to determine
an equation of exponential type satis…ed by the exponential e q!( k t 1=r ) . Doing that, we prove, in
the case when q! is a monomial, that the majors maj e q!( k t 1=r ) express themselves in terms of
the hypergeometric functions
pFq(b1; b2; :::; bp; c1; c2; :::; cq; t) = X (b1)m(b2)m:::(bp)m t
(c1)m(c2)m:::(cq)m
m
m!
or
0Fq( ; c1; c2; :::; cq; t) = X
m 0
m 0
1
(c1)m(c2)m:::(cq)m
t m
m!
; 1 p q
where b1; b2; :::; bp 2 C, c1; c2; :::; cq 2 Cn( N) and, for all 2 C, ( )m is the Pochhammer symbol:
( )0 = 1 and ( )m = ( + 1):::( + m 1) for all m 1
Proposition 4.15 ([Re07]) Let a 2 C be, r 2 and k 2 f1; :::; r 1g such that k
r
36
is irreducible.

at 1=2
1. When r = 2, the major maj e
at 1=2
maj e
with suitable 1; 2 2 C.
( ) = 1
at 1=3
2. When r = 3, the majors maj e
at 1=3
maj e
at 2=3
maj e
( ) = 1
( ) = 1
0F1
1F1
with suitable 1; 2; 3; 1; 2; 3 2 C.
at k=r
3. When r 4, the major maj e
at k=r
maj e
( ) =
Xr
1
j=1
reads
4 +
0
@ 2
2
3=2 e a 2
; 2
3 ; a3 27
0F1
+ 4=3 2
5
6
2 4a3 ; ; 3 27 2 1F1
+ 5=3 2
at 2=3
and maj e
reads
7
6
; 4
3 ; a3 27
1F1
2
a
read
5=3 + 3
1 3 a2 ; ; 2 2 4
0
@
4 4a3 ; ; 3 27 2 2F2
+ 7=3 3
2
9 3a3
2
1
2
; 1; 1
2
k 1Fr 2 b1;j; b2;j; :::; bk 1;j; c1;j; c2;j; :::; cr 2;j;
j
1+jk=r
e a2
4
1
A
1F2 1; 4 5 ; 3 3 ; a3 27
2 4a3 ; ; 3 27 2
k
kFr 1 b1; b2; :::; bk; c1; c2; :::; cr 1; k
+ r
with suitable 1; 2; :::; r 2 C and suitable parameters b1;j; :::; bk 1;j; b1; :::; bk 2 C, c1;j; :::; cr 2;j;
c1; :::; cr 1 2 Cn( N) and k;j; k 2 C.
In the case when q! is no more a monomial, the majors maj e q!( k t 1=r ) do not seem to
read in terms of special functions. For example, the exponential e t 2=3 +t 1=3
di¤erential equation
(216t 5
81t 6 )y 000 + (1080t 4
can maj e t 2=3 +t 1=3
216 3 by 000 + (64 144 + 1728 2
324t 5 )y 00
(144t 2
k;j
k
being solution of the
930t 3 + 180t 4 )y 0 + (64 64t 21t 2 )y = 0;
is a microsolution at 0 of the Borel transform equation
81 3 )by 00
(352 3090 + 324 2 )by 0 + (909 180 )by = 0
All solutions of this equation are holomorphic on C1 with an irregular singularity at 0. Thus, a
basis (by1; by2; by3) of solutions is also a basis of microsolutions at 0 and the major maj e t 2=3 +t 1=3
reads
maj e t 2=3 +t 1=3
= 1by1 + 2by2 + 3by3
with suitable 1; 2; 3 2 C. The solutions by1, by2 and by3 being not “classical”functions, it is the
same for maj et 2=3 +t 1=3
.
37
2
1
A

5 Stokes multipliers and structure of singularities
5.1 Stokes matrices and rank reduction
We use the terminology left-right in C with the following meaning: a point moves to the left when
its argument moves counterclockwise and to the right when it moves clockwise. We choose all
arguments in ] 2 ; 0]. This choice corresponds to the usual choice [0; 2 [ at in…nity. We choose
also the determination of log x such that log x is a real number when x > 0.
The anti-Stokes directions of System (1) associated with the …rst block of columns e f of e F
are the directions of maximal decay of the exponentials e qj(1=x) with qj 6= 0. To each non-zero
polynomial qj correspond the r anti-Stokes directions
k := arg(aj;r)
r
2k
r
2
2(k + 1)
r
;
2k
r
; k = 0; :::; r 1
regularly distribued around the origin x = 0.
Fix such a collection of anti-Stokes directions and consider, for " > 0 small enough, the natural
realizations
Y k "(x) = F k "(x)x L e Q(1=x)
of e Y (x) in the directions k ", where F k "(x) denotes the uniquely determined r-sum of e F (x) at
k ". The Stokes matrix In + C associated with k e Y in the direction k is the unipotent matrix
characterized by the relation
Y = Y + (In + C )
k k
k
where the sums Y and Y + to the right and to the left of k respectively are the analytic continu-
k
k
ation of Y k " and Y k+" respectively to the sector with vertex 0, bisecting direction k and opening
. Note that such analytic continuations exist without any ambiguity when " > 0 is small enough.
r
For all k = 0; :::; r 1, denote by c the …rst n1 columns of C and split c into J row-blocks
k k k
c j;
k
accordingly the Jordan structure of L. The entries of cj;
k are zero as soon as eqj(1=x) is not ‡at
in the direction k. We refer to such entries as trivial entries of c k . The other entries of c k are
called the Stokes multipliers of b Y associated with b f in the direction k.
The rank reduction allows to collect the r directions k into an only direction
= r 0 2] 2 ; 0]
that is an anti-Stokes direction of System (10) associated with e f. Using similar notations to those
seen above, the Stokes matrix Irn + C associated with e Y in the direction is the unipotent
matrix characterized by the relation (see [L-R01, Proposition 4.2])
Mr
1
Y = Y +(Irn + C ) where C = C k
38
k=0

which reads also
(42) s ( e F ) s +( e F ) = s +( e F )Y 0C Y 1
0
where s ( e F ) and s +( e F ) are the sums of e F to the right and to the left of respectively. Recall
(cf. Section 1.2) that
s ( e Z 1ei F )(t) := bF ( )e
=t
d
0
The aim of Section 5 is to make explicit the Stokes multipliers associated with the …rst block of
columns e f of e F (initial System (1)) in terms of resurgence multipliers (the resurgent coe¢ cients that
are pertinent) associated with b f (r-reduced System (10)) and, reciprocally, to make explicit the
structure of singularities of b f from the formal fundamental solution e Y and the Stokes multipliers
associated with e f. These formulae allow us, as in the case of systems with single level 1 (see
[L-RR]), to state the existence of a linear isomorphism between the full set of Stokes multipliers of
the initial System (1) and the full set of the resurgence multipliers of the r-reduced System (10) .
5.2 Main result
Denote by
d the half-line d := [0; 1e i [ issuing from 0 toward 2 C ;
( b f) = faj;r 6= 0 ; arg(aj;r) = g the set of singularities of b f on the half-line d ;
J! = fj 2 f1; :::; Jg ; aj;r = !g for all singularity ! 2 ( b f);
0p q the null matrix of size p q;
c j;q
k
the q-th column of cj;
k ;
= e 2i =r and its conjugate in C.
The resurgence coe¢ cients of b f at any singularity ! 2 ( b f) depend on the path chosen for
the analytic continuation of b f. We always choose the analytic continuation of b f to the right of all
singularities met along d between 0 and ! like shown in Figure 3.
Figure 3
39

Since the set ( b f) is …nite and e f summable-resurgent, we can apply Ecalle’s method detailed
in [L-RR, Section 4] to Identity (42). We obtain thus, for all q 2 f1; :::; n1g, the identities
X
(43)
e !=t
Z
! b f ;q
=t 1 X
( )e d = s +(
r
e F )M ;q
! (t)e !=t
!2 ( b f)
+
!2 ( b f)
where + is the Hankel loop around 0 like shown in Figure 2 and M ;q
! is the rn-dimensional
column-vector de…ned, for u = 1; :::; r and j = 1; :::; J, by
8
0nj 1 if j =2 J!
(44) M u;j;q
! (t) =
><
>:
Xr
1
k=0
( kt1=r ) Lj (u 1)Inj c j;q j;q 1
c k k
log( k t 1=r )+
::: + ( 1)q 1 log q 1 ( kt1=r )
c
(q 1)!
j;1
e k q!( kt 1=r ) if j 2 J!
The global identity (43) splits into identities term by term as proved below:
Theorem 5.1 For all singularity ! 2 ( b f), the Laplace transform
L +( ! b f ;q
Z
)(t) := ! b f ;q
=t
( )e d
of ! b f ;q
is connected to the Stokes multipliers associated with e f by the relation
(45) L +( ! b f ;q
)(t) = 1
r s +( e F )M ;q
! (t)
The vectors M ;q
! (t) are those de…ned above.
Proof. The proof is straightforward from Lemmas 5.2 and 5.3 proved just below.
Indeed, due to the description of singularities given in Theorems 4.2 and 4.14, we deduce from
Lemma 5.2 an explicit formula for the Laplace integrals L +( ! b f ;q
). Precisely,
when ! 2 ( b f) is with monomial front (cf. Theorem 4.2), L +( ! b f ;q
) reads
(46) L +( ! b f ;q
N!;0+q
X
)(t) =
p=0
+
X
;q
!;p log p N!;0+q
t +
p=1
s +( e k ;q
!;p) log p t
+ X
2 !
N!; +q
X
where ;q
!;p 2 Mrn;1(C), e k ;q
!;p 2 Mrn;1(tCftg 1; +) and e K ;q
!; ;p 2 Mrn;1(Cftg 1; +);
40
p=0
s +( e K ;q
!; ;p ) logp t t

when ! 2 ( b f) is with non-monomial front (cf. Theorem 4.14), L +( ! b f ;q
) reads
L +( ! b f ;q
% 1 X
Z
)(t) =
=t
d
!
e q!( kt 1=r )
k=0
k=0
then
(47) L +( ! b f ;q
0
% 1 X
B
X
)(t) = @
where 2 C, s 2 N and
;s
…nite
+
' ;q
!;k ( )e
s +( e ;q
e ;q
!;k; ;s 2 Mrn;1(Cftg 1; +).
1
C
!;k; ;s )t logs tA
e q!( kt 1=r )
On the other hand, it results from (44) that the matrix product 1
r s +( e F )M ;q
! (t) reads similarly
to (46) when ! is with monomial front and similarly to (47) when ! is with non-monomial front.
Indeed,
when ! 2 ( b f) is with monomial front, condition q! 0 implies that all exponentials
e q!( k t 1=r ) are equal to 1;
when ! 2 ( b f) is with non-monomial front, Lemma 4.13 implies the decomposition
(48) M ;q
% 1 X
! (t) = M ;q
!;k (t)eq!( kt 1=r )
k=0
where, for all k 2 f0; :::; % 1g, M ;q
u = 1; :::; r and j = 1; :::; J, by
!;k is the rn-dimensional column-vector de…ned, for
8
0nj 1 if j =2 J!
M u;j;q
!;k (t) =
><
>:
X1
`=0
( k+`% t1=r ) Lj (u 1)Inj c j;q j;q 1
c k+`%
k+`% log( k+`% t 1=r )+
::: + ( 1)q 1 log q 1 ( k+`% t1=r )
c
(q 1)!
j;1
k+`%
if j 2 J!
Therefore, we deduce from Lemma 5.3 that one can identify the exponentials e !=t on both sides
of identity (43).
Lemma 5.2 Let e' be a 1-Gevrey series and b' its minor:
b'( ) = X
m 0
'm m 2 Cf g
41

Given 2 C and ` 2 N, we denote by
e' ;`(t) := 2i X
For all p 2 N, the function b'( )
e (t) :=
pX
k=0
m 0
d `
dz `
1 log p
i z e
(1 z) jz=m+
is a major of
Moreover, if e' is summable in the direction + , then
'mt m 2 C[[t]]1
p
k e' ;p k(t)t log k t 2 t C[[t]]1[log t]
1. e' ;` is summable in the direction + for all 2 C and ` 2 N;
2. for all p 2 N,
Proof. Recall the identity
Z
(49)
Z
+
b'( )
+
1 log p e
1 e
=t d =
=t d = 2i e i
and more generally, for all p 2 N, the identity
(50)
Z
+
1 log p e
=t d = 2i
pX
k=0
p
k
pX
k=0
p
k s +(e' ;p k)(t)t log k t
t ; 2 C
(1 )
p d k
dzp k
i z e
(1 z) jz=
obtained by derivating p times with respect to . The function
'(t) := 2i
pX
k=0
p
k
p d k
dzp k
being a resurgence constant, Section 1.3.1 implies that
i z e
(1 z) jz=
1 log p
Consider the case p = 0 and prove the identity
Z
1 =t
(51)
b'( ) e d = s +(e' ;0)(t)t
+
t log k t ; 2 C
t log k t
is a major of '(t).
The case when 2 Z being clear by the residue theorem (e' ;0 is polynomial since 1
( m)
= 0 for
all m 2 N), we assume 2 CnZ. From (49), it results that b'( ) 1 is a major of e' ;0(t)t . Since
the variation of b'( ) 1 is
var b'( )
1 ( ) = 1 e 2i
42
b'( )
1

the summability of e' implies that e' ;0(t)t 2 H + ;exp. Thus, Proposition 1.6 applies given the
summability of e' ;0 in the direction + and Identity (51).
A recursion allows to extend Identity (51) to any integer p
a major of
1. Indeed, b'( ) 1 p
log being
pX
e (t) =
p
k e' ;p k(t)t log k t
k=0
(cf. (50)), the summability of e' implies once more that e 2 H + ;exp. On the other hand, the
e' ;p k’s being summable in the direction + for all k = 1; :::; p, Proposition 1.5 shows that the
e' ;p k(t)t log k t’s belong also to H + ;exp for k = 1; :::; p. Therefore e' ;p(t)t 2 H + ;exp and Proposition
1.6 applies again.
Lemma 5.3 Let m 2 be an integer and '1, ..., 'm the functions de…ned on a sector with
vertex 0, bisecting direction and larger than a half-plane by
pj X
'j(t) :=
`=1
s +( e hj;`)t ` log r` t e p`(t 1=r )
where e hj;` 2 Cftg 1; +[t 1 ], ( `; r`) 2 C=Z N distinct and p`(t) 2 tC[t] of degree not greater than
r 1 and distinct two by two.
Assume that the 'j’s satisfy on an identity of the form
'1 + '2e !2=t + ::: + 'me !m=t = 0
where arg(!j) = for all j = 2; :::; m and j!2j < j!3j < ::: < j!mj.
Then, 'j 0 for all j 2 f1; :::; mg.
Proof. The proof is similar to the one of [L-RR] and is left to the reader.
Theorem 5.1 allows to state several results on the connections between the Stokes multipliers
and the structure of singularities in the Borel plane. Although these results were similar for all
kinds of singularities, we prefer to distinguisch the case of singularities with monomial front and
the case of singularities with non-monomial front.
5.3 Connections between Stokes multipliers and structure of singularities
with monomial front
A …rst consequence of Theorem 5.1 is to make explicit the analytic continuation of e f to the right
of its singularities with monomial front according to the formal fundamental solution e Y and the
Stokes multipliers associated with e f:
43

Corollary 5.4 (structure of singularities with monomial front)
For all ! 2 ( b f) with monomial front, ! b f ;q
is a major of the matrix product 1
r e F M ;q
! .
Reciprocally, knowing the structure of singularities with monomial front in the Borel plane, we
can determine the Stokes multipliers associated with e f and non-zero monomials qj’s:
Corollary 5.5 (Stokes versus resurgence multipliers)
Let q 2 f1; :::; n1g be. Given ! 2 ( b f) a singularity of b f with monomial front (cf. Theorem 4.2),
we consider b f ;q
the analytic continuation of the q-th column of b f to the right of ! along d .
For all j 2 J!, the matrices c j;q
, k = 0; :::; r 1, are the unique solutions of the blocked system
(52)
where
2
6
1. Cj;q := 6
4
8
><
>:
2
Inj
6
4 .
Inj
Inj
Cj;q;1
Cj;q;2
.
Cj;q;r
3
k
Lj (r 1)Lj
Lj In j (r 1)(Lj In j)
.
.. .
Lj (r 1)In j (r 1)(Lj (r 1)In j)
.
3 2
c
7 6
7 6
7 6
5 4
j;q
0
c j;q
1
.
c j;q
r 1
7
5 is the rnj-dimensional column-vector de…ned by
Cj;q;1 :=
Cj;q;u :=
N!;0+q
X
p=0
N
j
!;
r
X
p=0
3
7
5 = rCj;q Xj;q
p(0) 1;j;q
!;p if j = 0
+q
u 1
r
p
j
r
u 1
r
H u;j;q
!; j
r
u 1
r ;p(0) if j 6= 0 or u 6= 1
where the complex numbers p( ) are de…ned, for p 2 N and 2 C, by
p( ) := 2i dp
dz p
44
i z e
(1 z) jz=

2. Xj;q is the rnj-dimensional column-vector de…ned by
2
where, for …xed k 2 f0; :::; r 1g,
8
Xj;1;k = 0
><
>:
and, for q 2,
j;q 1
Xj;q;k = (k log )c
6
Xj;q := 6 r 1
4
X
k=0
Xr
1
k=0
Xr
1
k=0
kLj Xj;q;k
k(Lj In j) Xj;q;k
.
k(Lj (r 1)In j) Xj;q;k
k + ::: + ( 1)` (k log ) `
`!
j;q `
c
Proof. System (52) results from the identi…cation of the terms in t j
r
3
7
5
k + ::: + ( 1)q 1 (k log )
(q 1)!
j 2 J!, in the identity deduced from (45) by asymptotic expansion at the origin.
The uniqueness is provided by the fact that the matrix
2
3
is invertible 6 .
Inj
6
4 .
Inj
Inj
Lj (r 1)Lj
Lj In j (r 1)(Lj In j)
.
.. .
Lj (r 1)In j (r 1)(Lj (r 1)In j)
.
7
5
q 1
c j;1
k
u 1
r , u 2 f1; :::; rg and
These two corollaries allow to state the existence of a linear isomorphism between the Stokes
and resurgence multipliers associated with singularities with monomial front:
Proposition 5.6 (linear isomorphism)
Given ! 2 ( b f) a singularity of b f with monomial front (cf. Theorem 4.2), we consider the
analytic continuation of b f to the right of ! along d .
For all j 2 J!, the Laplace transformation L + induces a linear isomorphism between
6 Its inverse is
1
r
2
6
4
Inj Inj Inj
Lj Lj In j Lj (r 1)In j
.
.
(r 1)Lj (r 1)(Lj In j ) (r 1)(Lj (r 1)In j )
45
. ..
.
3
7
5

1. (case j = 0) the entries of c j;
u 2 f2; :::; rg;
2. (case j 6= 0) the entries of c j;
k k=0;:::;r 1
k k=0;:::;r 1
and the entries of
and the entries of Hu;j;
!; j
r
1;j;
!;0 and H u;j;
u 1
!; ;0(0)
with
r
with u 2 f1; :::; rg.
u 1 ;0(0) r
Proof. Corollary 5.4 implies that, for all j 2 J! and q 2 f1; :::; n1g, the entries of 1;j;q
!;p (resp.
H u;j;q
!; j are linear combinations in C of the entries of
u 1 ;p(0)) r r
1;j;s
!;0 (resp. H u;j;s
!; j
s 2 f1; :::; qg. The result follows easily by Corollary 5.5.
r
De…nition 5.7 The entries of the matrices
are called resurgence multipliers of b f at !.
1;j;
!;0 and H u;j;
!; j
r
with
u 1 ;0(0)) r
u 1
r ;0(0), j 2 J!, of Proposition 5.6
By misuse of language, we also call resurgence multipliers the entries of the matrices 1;j;
!;p and
H u;j;
!; j
r
u 1
r ;p(0), j 2 J! and p 0, of Corollary 5.5. The entries of their q-th column de…ne the
resurgence multipliers of b f ;q
at !.
The case when L is diagonal deserves to be precised since, contrary to the general case, it gives
explicitly the linear isomorphism between Stokes and resurgence multipliers:
Corollary 5.8 Assume that the matrix L is diagonal: L = diag(0; 2; :::; n).
Given ! 2 ( b f) a singularity of b f with monomial front (cf. Corollary 4.4), we consider b f the
analytic continuation of b f to the right of ! along d .
For all j 2 J!, the Stokes multipliers c j
1;j
! and H u;j
!; j
r
u 1
r
1. case j = 0:
c j
(0) by the following formulae:
k = 2i 1;j
! + 2i
1;j
! = 1
H u;j
!;
Xr
1
c
2 ir
j
k=0
u 1 (0) =
r
k
rX
u=2
, k = 0; :::; r 1, are related to the resurgence multipliers
k
i (u 1)
k(u 1) e r
(1 +
i
u 1 (1 + )e r
(u
r
1)
Xr
1
2 ir
k=0
u 1
r )Hu;j
!;
46
k(u 1) c j
u 1 (0) for all k 2 f0; :::; r 1g
r
k
for all u 2 f2; :::; rg

2. case j 6= 0:
c j
k
H u;j
!; j
r
= 2i
rX
u=1
u (0) = 1
r
(1
k( j u+1) e i ( j u+1)
r
(1
2 ir
j u+1
r
) Hu;j
!; j
r
j u+1
)e r i ( j u+1)
r 1
r X
k=0
u (0) 1
for all k 2 f0; :::; r 1g
r
k( j u+1) c j
k for all u 2 f1; :::; r 1g
5.4 Connections between Stokes multipliers and structure of singularities
with non-monomial front
To each result previously enunciated in the case of singularities with monomial front corresponds
an analogous result in the case of singularities with non-monomial front. Their proofs being similar
to those that have already been made, we give only their terms.
Just as in the case of singularities with monomial front, Theorem 5.1 provides the structure
of singularities with non-monomial front in terms of the formal fundamental solution e Y and the
Stokes multipliers associated with e f:
Corollary 5.9 (structure of singularities with non-monomial front)
For all ! 2 ( b f) with non-monomial front, ! b f ;q
is a major of the matrix product 1
r e F M ;q
! .
In particular, Decomposition (48) of the matrix M ;q
! allows, due to Proposition 1.3, to improve
Theorem 4.14 given the precise possible form of singularities with non-monomial front:
Corollary 5.10 (description of singularities with non-monomial front)
Let ! be any of aj;r 6= 0 with non-monomial front and q 2 f1; :::; n1g. The q-th column b f ;q
translated by ! reads, modulo germs of analytic function at 0:
! b f ;q
% 1 X
( ) =
where, for all k 2 f0; :::; % 1g, ' ;q
!;k
k=0
' ;q
!;k ~ maj
rami…ed singularity at 0.
Precisely, modulo germs of analytic function at 0, ' ;q
!;k reads
' ;q
N!;0+q
X
!;k ( ) =
p=0
;q
N!;0+q+1
X
!;k;p p
log +
p=1
e q!
1
k t 1=r
!!
( )
1
is a major of
r e F M ;q
;q
!;k . In particular, ' !;k
h ;q
!;k;p ( ) logp + X
47
2 !
N!; +q
X
p=0
H ;q
!;k; ;p ( ) logp 1
of b f
has a simple

;q
where !;k;p 2 Mrn;1(C) and h ;q
;q
!;k;p ( ); H !;k; ;p ( ) 2 Mrn;1(Cf g). The set of rami…cation exponents
! and the logarithmic orders (N!;0 + q; (N!; + q) 2 !) are given by
8
j u 1
! =
; j 2 f1; :::; Jg and u 2 f1; :::; rg such that
r r
><
>:
N!;0 = max (nj 2 ; j 2 f1; :::; Jg such that qj 2 F r(!) and j = 0)
N!; = max (nj 2 ; j 2 f1; :::; Jg such that qj 2 F r(!) and
there exists u 2 f1; :::; rg such that
qj 2 F r(!) and ( j 6= 0 or u 6= 1)g
j
r
u 1
r =
Moreover, the functions h ;q
;q
!;k;p and H !;k; ;p are germs of endless continuable analytic function with
singular support the points aj;r ! that grow exponentially at in…nity.
;q
These functions and the constants !;k;p are so-called resurgence coe¢ cients of b f ;q
at !.
These two corollaries allow to state results similar to Corollary 5.5 and Proposition 5.6 concerning,
on the one hand, the determination of Stokes multipliers from resurgence multipliers and, on
the other hand, the existence of a linear isomorphism between Stokes and resurgence multipliers:
Corollary 5.11 (Stokes versus resurgence multipliers)
Let q 2 f1; :::; n1g be and k 2 f0; :::; % 1g. Given ! 2 ( b f) a singularity of b f with non-monomial
front (cf. Corollary 5.10), we consider b f ;q
the analytic continuation of the q-th column of b f to
the right of ! along d :
For all j 2 J!, the matrices c j;q
, ` = 0; :::; 1, are the unique solutions of the blocked system
2
6
4
where
k+`%
kLj (k+%)Lj (k+( 1)%)Lj
k(Lj In j ) (k+%)(Lj In j ) (k+( 1)%)(Lj In j )
.
.
k(Lj ( 1)In j ) (k+%)(Lj ( 1)In j ) (k+( 1)%)(Lj ( 1)In j )
1. the matrix
2
6
4
.. .
.
3 2
7 6
7 6
7 6
5 4
kLj (k+%)Lj (k+( 1)%)Lj
k(Lj In j ) (k+%)(Lj In j ) (k+( 1)%)(Lj In j )
.
.
. ..
3
c j;q
k
c j;q
k+%
.
c j;q
7
5
k+( 1)%
= rCk;j;q Xk;j;q
k(Lj ( 1)In j ) (k+%)(Lj ( 1)In j ) (k+( 1)%)(Lj ( 1)In j )
48
.
3
7
5

is invertible with inverse
2
2
6
2. Ck;j;q := 6
4
8
><
>:
1
Ck;j;q;1
Ck;j;q;2
.
Ck;j;q;
6
4
Ck;j;q;1 :=
Ck;j;q;u :=
3
kLj k(Lj In j ) k(Lj ( 1)In j )
(k+%)Lj (k+%)(Lj In j ) (k+%)(Lj ( 1)In j )
.
.
(k+( 1)%)Lj (k+( 1)%)(Lj In j ) (k+( 1)%)(Lj ( 1)In j )
.. .
7
5 is the nj-dimensional column-vector de…ned by
N!;0+q
X
p=0
N
j
!;
r
X
p=0
p(0) 1;j;q
!;k;p if j = 0
+q
u 1
r
p
j
r
u 1
r
H u;j;q
!;k; j
r
.
3
7
5
u 1
r ;p(0) if j 6= 0 or u 6= 1
where the complex numbers p ( ) are de…ned, for p 2 N and 2 C, by
p( ) := 2i dp
dz p
3. Xk;j;q is the nj-dimensional column vector de…ned by
2
6
Xk;j;q := 6 1
4
X
where, for …xed ` 2 f0; :::; 1g,
8
Xk;j;1;` = 0
><
>:
`=0
X1
`=0
and, for q 2,
q 1 X
Xk;j;q;` =
p=1
X1
`=0
i z e
(1 z) jz=
(k+`%)Lj Xk;j;q;`
(k+`%)(Lj In j) Xk;j;q;`
.
(k+`%)(Lj ( 1)In j) Xk;j;q;`
( 1) p ((k + `%) log ) p
49
p!
3
7
5
j;q p
c k+`%

Proposition 5.12 (linear isomorphism)
Given ! 2 ( b f) a singularity of b f with non-monomial front (cf. Corollary 5.10), we consider the
analytic continuation of b f to the right of ! along d .
For all j 2 J! and k 2 f0; :::; % 1g, the Laplace transformation L + induces a linear isomorphism
between
1. (case j = 0) the entries of c j;
u 2 f2; :::; g;
k+`%
2. (case j 6= 0) the entries of c j;
f1; :::; g.
k+`%
`=0;:::; 1
`=0;:::; 1
and the entries of
and the entries of H u;j;
1;j;
!;k;0 and Hu;j;
!;k;
!;k; j
r
u 1 ;0(0)
with
r
with u 2
u 1 ;0(0) r
Note that these results, and especially Corollary 5.11, are rather theoretical results. It does
not seem that these ones allow the e¤ective calculation of the Stokes multipliers associated with
ef and singularities with non-monomial front. Indeed, the study of the analytic continuation of b f
near any of its singularities ! with non-monomial front does not allow to determine, in general,
the resurgence multipliers at ! (see, e.g., Section 4.2.1). To circumvent this di¢ culty, we can
proceed like in [L-R90] by …rst performing the singularity ! into a singularity with monomial front
by means of a suitable algebraic change of the variable x in the initial System (1), then applying
Corollary 5.6. See Section 6.3 below for an example.
6 Examples
We complete this paper by three simple examples where we can calculate the exact value of Stokes
multipliers from the study of the analytic continuation in the Borel plane. Of course, these examples
are not general, but they suitably illustrate the use of previous results.
6.1 Example 1
In this …rst example, the matrix L is diagonal and b f admits two collinear singularities with
monomial front.
Consider the system
3 dY
(53) x
dx =
2
6
4
0 0 0
x 4 x 5 2 0
x 4 + x 5 0 4 + x2
2
50
3
7
5 Y

with single level 2 and its formal fundamental solution e Y (x) = e F (x)xLeQ(1=x) de…ned by
2
1
eF (x) = 4
0
1
0
3
0
05
= I3 + O(x
1
4 ) ; L = diag 0; 0; 1
2
and Q 1
x
= diag 0;
1
;
x2 The …rst column e f of e F has two anti-Stokes direction ( = 0 and = ) determined by
the directions of maximal decay of e 1=x2 and e 2=x2.
The Stokes matrices I3 + C0 and I3 + C
associated with = 0 and = respectively are de…ned by
2
0 0
3
0
2
0 0
3
0
C0 = 4 5 and C = 4 5
c 2 0 0 0
c 3 0 0 0
c 2 0 0
c 3 0 0
Indeed, only the …rst column of e F is divergent. The Stokes multipliers c 2 0 and c 2 on the one hand
and c 3 0 and c 3 on the other hand are calculated from the resurgence multipliers of b f at 1 and 2
respectively by means of Corollary 5.8 (these two singularities are with monomial front and L is
diagonal).
The 2-reduced system of System (53) is de…ned by
2
2 dY
2t
dt =
6
4
0 0 0 0 0 0
t2 2 0 t3 0 0
t2 0 4 + t
2 t3 0 0 0 t
0
0
0
0
t2 0 0 t2 2 t 0
t2 0 0 t2 0 4 t
2
Consequently, the formal series e f reads (cf. System (17))
2
1
6
6ef
6
ef = 6
4
1;2
(t)
ef 1;3
(t)
0
ef 2;2
(t)
ef 2;3
3
7
5
(t)
where the formal series e f u;j
, u = 1; 2 and j = 2; 3, satisfy e f u;j
(t) = O(t 2 ) and the di¤erential
51
3
7 Y
7
5
2
x 2

system
8
><
>:
2t 2 de f 1;2
dt
2t 2 de f 1;3
dt
2t 2 de f 2;2
dt
2t 2 de f 2;3
dt
2e f 1;2
= t2 4 + t
2 e f 1;3
= t 2
(2 t) e f 2;2
= t2 4
t
2 e f 2;3
= t 2
By Borel transformation, the minor b f reads
2
2 3 0
1 6
6
607
6bf
7 6
6
bf( ) = 607
6
7
6
607
+ 6
7 6
405
6
4
0
1;2
( )
bf 1;3
( )
0
bf 2;2
( )
bf 2;3
3
7
5
( )
where b f u;j
, u = 1; 2 and j = 2; 3, satisfy b f u;j
(0) = 0 and the di¤erential system
Therefore,
8
><
>:
2( 1) b f 1;2
=
2( 2) db f 1;3
d
+ 3
2 b f 1;3
= 1
2( 1) db f 2;2
d + 3b f 2;2
= 1
2( 2) db f 2;3
d
+ 5
2 b f 2;3
= 1
bf 1;2
and b f 2;2
have only one singularity, located at = 1, and their analytic continuations to
the right of 1 along d0 = R + are given by
bf 1;2
( ) = 2( 1)
and
52
b f 2;2
( ) = i
3 ( 1) 3=2 1
3

f 1;3
and b f 2;3
have only one singularity, located at = 2, and their analytic continuations to
the right of 2 along d0 are given by
bf 1;3
( ) = 27=4
3 e 3i =4 ( 2) 3=4 + 2
3
We deduce thus from Corollary 5.8 that
8
><
>:
c 2 0 =
c 3 0 = 2 3=4
6.2 Example 2
4 p
3
4
3 ( 3
4
and
i , c 2 = + 4p
3
16
+
) 5
( 3
4 ) i , c3 = 2 3=4 4
3 ( 3
4 )
2;3
fb ( ) = 29=4
5 e 5i =4 ( 2) 5=4 + 2
5
In this second example, the matrix L has a 2-dimensional Jordan block and b f has only one
singularity that is with monomial front.
Consider the system
3 dY
(54) x
dx =
2
4
0 0
3
0
5 Y
x 4 + x 5 2 x 2
x 4 + 2x 5 0 2
with single level 2 and its formal fundamental solution e Y (x) = e F (x)xLeQ(1=x) de…ned by
2
1
eF (x) = 4
0
1
0
3
0
05
= I3 + O(x
1
4 ) ;
2
0
L = 40
0
0
0
0
3
0
15
0
and Q 1
x
= diag 0;
1
;
x2 The anti-Stokes directions of the …rst column e f of e F are the directions = 0 and = (the
directions of maximal decay of e 1=x2)
and the Stokes matrices in these directions are I3 + C0 and
I3 + C respectively where
2 3
2 3
C0 =
The four Stokes multipliers c 2;1;1
0
0 0 0
4c
2;1;1
0 0 05
and C = 4
c 2;2;1
0 0 0
i
0 0 0
5
c 2;1;1 0 0
c 2;2;1 0 0
16
5
( 3
4 )
, c 2;2;1
0 , c 2;1;1 and c 2;2;1 are calculated from the resurgence multipliers
of b f at 1 by means of Corollary 5.5 ( = 1 is a singularity with monomial front and L is not
diagonal).
53
1
x 2

The 2-reduced system of System (54) is de…ned by
2
2 dY
2t
dt =
6
4
0 0 0 0 0 0
t 2 2 t t 3 0 0
t 2 0 2 2t 3 0 0
0 0 0 t 0 0
t 2 0 0 t 2 2 t t
2t 2 0 0 t 2 0 2 t
The formal series e f reads therefore (cf. System (17))
2
1
6
6ef
6
ef(t) = 6
4
1;2;1;1
(t)
ef 1;2;2;1
(t)
0
ef 2;2;1;1
(t)
ef 2;2;2;1
3
7
5
(t)
where e f u;2;`;1
, u = 1; 2 and ` = 1; 2, satisfy e f u;2;`;1
(t) = O(t2 ) and the di¤erential system
8
2t2 de f 1;2;1;1
dt
2e f 1;2;2;1
te f 1;2;2;1
= t2 Thus, the minor b f reads
><
>:
2t 2 de f 1;2;2;1
dt
2t 2 de f 2;2;1;1
dt
2t 2 de f 2;2;2;1
dt
bf( ) =
2e f 1;2;2;1
= t2 (2 t) e f 2;2;1;1
(2 t) e f 2;2;2;1
= 2t2 2
2 3 0
1 6
6
607
6bf
7 6
6
607
6
7
6
607
+ 6
7 6
405
6
4
0
1;2;1;1
( )
bf 1;2;2;1
( )
0
bf 2;2;1;1
( )
bf 2;2;2;1
3
7
5
( )
54
3
7 Y
7
5
te f 2;2;2;1
= t2

where b f u;2;`;1
, u = 1; 2 and ` = 1; 2, satisfy b f u;2;`;1
(0) = 0 and the identities
8
><
>:
2( 1) b f 1;2;1;1
= + 1 b f 1;2;2;1
2( 1) b f 1;2;2;1
=
2( 1) db f 2;2;1;1
d
2( 1) db f 2;2;2;1
d
+ 3b f 2;2;1;1
= 1 + b f 2;2;2;1
+ 3b f 2;2;2;1
= 2
Solving these equations by the method of variation of constants, we deduce that the analytic
continuations of the b f u;2;`;1
’s to the right of 1 along d0 = R + read
8
><
>:
bf 1;2;1;1
( ) =
bf 1;2;2;1
( ) = 2( 1)
bf 2;2;1;1
( ) =
3 + i log( 1)
+
4( 1) 4( 1)
3 5i
( 1)
9
3=2 i
3 ( 1) 3=2 log( 1) + 5
9
bf 2;2;2;1
( ) = 2i
3 ( 1) 3=2 + 2
3
Apply now Corollary 5.5: Identity (52) implies
2
1
6
60
41
0
0
1
0
1
1
0
1
0
3 2
i c
1 7 6
7 6
i 5 4
1
2;1;1
0
c 2;2;1
0
c 2;1;1
c 2;2;1
3
7
5 =
2
3 + i
6
63
6
4
4
5i
0
9
55
0(0) + 1
4 1(0)
1
2 0(0)
1
2
2i
3 0
1
2
i
3 1
1
2
3
7
5

1
where the complex numbers p(0) and p( ), p = 0; 1, are de…ned by
2
8
><
>:
Therefore,
0(0) = 2i
1(0) = 2i d
dz
0
1
8
><
>:
1
2
1
2
c 2;1;1
0
c 2;2;1
0
c 2;1;1 =
6.3 Example 3
= 2i ei =2
= 2i d
dz
= ip
18
i z e
= 2
(1 z) jz=0
2 2i ( is Euler’s constant)
( 3 = 4p
) 2
e
i z
(88 + 27p
= ip
3 (8 + 3p )
8 p
(1 z) jz= 1
2
24 9 p
= p (4 8 + 8 log 2 + 4i )
48 log 2)
3 + 2 + ip
18 (88 + 27p + 24 9 p + 48 log 2)
c 2;2;1 = ip
3 (8 3p )
In this last example, we treat in detail the case of a singularity with non-monomial front. Let
3 dY
(55) x
dx
= 0 0
x 4 2 + x Y
be the system of Section 4.2.1. Recall that the formal fundamental solution e Y is de…ned by
eY (x) = e F (x)e Q(1=x) where
eF (x) =
1 0
1 = I2 + O(x 4 ) and Q 1
x
= diag 0;
The anti-Stokes directions of the …rst column e f of e F are once more = 0 and = (the
directions of maximal decay of e 1=x2 1=x ) and the associated Stokes matrices are de…ned by I2+C0
and I2 + C respectively where
C0 =
0 0
c 2 0 0
and C =
56
0 0
c 2 0
1
x 2
1
x

The singularity = 1 of b f being with non-monomial front, the calculation of the Stokes multipliers
is much more involved than in the previous examples. Indeed, the study of the analytic continuation
of b f near 1 (cf. Section 4.2.1) does not allow to calculate easily the resurgence multipliers and,
therefore, the Stokes multipliers.
To circumvent this di¢ culty, we perform the singularity = 1 into a singularity with monomial
front. Let us
System (55) becomes
3 dY
(56) y
dy =
x = y
1
2
4
y
2
3
0 0
5 Y
8y4 2
(2 y) 3
and the formal fundamental solution e Y(y) = e F1(y)e Q1(1=y) associated with e Y (x) is de…ned by
eF1(y) = e F (x(y))
1 0
0 e 1=4 =
1 0
0 e 1=4 + O(y 4 ) and Q1
1
y
= diag 0;
Systems (55) and (56) have the same Stokes matrices ([L-R90]). To normalize e F1 to e F1(0) = I2,
we use the constant gauge transformation
This one transforms System (56) into
Z =
3 dZ
(57) y
dy =
2
4
1 0
0 e 1=4 Y
3
0 0
5 Z
8e 1=4 y 4
(2 y) 3 2
and the formal fundamental solution e Y(y) into the formal fundamental solution e Z(y) = e F2(y)eQ1(1=y) where
eF2(y) =
1 0
0 e 1=4 e F1(y) = I2 + O(y 4 )
Systems (55) and (57) have once more the same Stokes matrices. The singularity = 1 being now
with monomial front, we can apply Corollary 5.8 (the matrix L is null).
The 2-reduced system of System (57) is de…ned by
2 dY
2t
dt =
2
0
6
6T1(t)
4 0
0
2
0
0
tT2(t)
t
0
0
0
3
7
5
T2(t) 0 T1(t) 2 t
Y
57
1
y 2

where 8> <
>:
The …rst column e f 2 of e F 2 reads
T1(t) = 16e 1=4 (4 + 3t)t2 64 48t + 12t2 = 16e 1=4X (2m 3)(m 1)
t3 4
m 2
m tm T2(t) = 8e 1=4 (12 + t)t2 64 48t + 12t2 = 8e 1=4X (2m 1)(m 1)
t3 4
m 2
m tm 2
1
6
ef
6ef
2(t) = 6
4
1;2
2 (t)
0
ef 2;2
3
7
5
2 (t)
where the formal series e f 1;2
2 and e f 2;2
2 satisfy e f 1;2
2 (t) = e f 2;2
2 (t) = O(t 2 ) and the di¤erential system
8
Thus, the minor b f 2 of e f 2 reads
><
>:
2t2 de f 1;2
2
dt
2t2 de f 2;2
2
dt
bf 2( ) =
2 e f 1;2
2 = T1(t)
(2 t) e f 2;2
2 = T2(t)
2 3
1
6
60
7
405
0
+
2
0
6
6bf
6
4
1;2
2 ( )
0
bf 2;2
3
7
5
2 ( )
where b f 1;2
2 and b f 2;2
2 satisfy b f 1;2
2 (0) = b f 2;2
2 (0) = 0 and the identities
with 8> <
>:
8
><
>:
2( 1) b f 1;2
2 = b T1( )
2( 1) db f 2;2
2
d + 3b f 2;2
bT1( ) = 16e 1=4X 2m 1
4m+1 (m 1)!
m 1
bT2( ) = 8e 1=4X 2m + 1
4m+1 (m 1)!
m 1
58
2 = d b T2
d
( )
m ( + 2) ( 1)=4
= e
2
m ( + 6) ( 1)=4
= e
4

Therefore the analytic continuations of b f 1;2
2 and b f 2;2
2 to the right of 1 along d0 = R + are de…ned
by 8> <
where
>:
bf 1;2
2 ( ) =
bf 2;2
2 ( ) = 1
2
= i
4
( + 2) 1)=4
e(
4( 1)
Z
3=2 ( 1)
0
e 1=4 + 3
2 1F1
We deduce thus from Corollary 5.8
References
8
><
>:
c 2 0 = ip
2
c 2 = ip
2
( 1) 1=2 d e T2
d ( )d = ( 1) 3=2 + Hol( 1)
3 p
1 3
;
2 2 ;
1
4
2e 1=4 31F1
3 p + 2e 1=4 + 31F1
and Hol( ) 2 Cf g
1 3
;
2 2 ;
1 3
;
2 2 ;
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60