arg z = „ ’ µ or vice versa. The set of all these directions, for arbi-

trary (n, m), are exactly the Stokes directions, i.e., the boundary rays

of the sectors Sj .

142 9. Stokes™ Phenomenon

4. Show that every half-open interval of length π/k contains the same

number of Stokes directions, say, n1 . For Suppj,n1 = Suppj+1 ∪ . . . ∪

Suppj+n1 and distinct n, m, ∈ { 1, . . . , µ }, show:

(a) (n, n) ∈ Suppj,n1 .

(b) (n, m) ∈ Suppj,n1 ⇐’ (m, n) ∈ Suppj,n1 .

(c) (n, m) ∈ Suppj,n1 and (m, ) ∈ Suppj,n1 ’ (n, ) ∈ Suppj,n1 .

(d) The matrices of the form I +C, with Cnm = 0 whenever (n, m) ∈

Suppj,n1 , form a group Gj,n1 with respect to multiplication.

(e) Statements (a)“(d), except for the backward direction in (b),

also hold for Suppj instead of Suppj,n1 . In the sequel, denote

the corresponding group by Gj .

(f) For j + 1 ¤ ¤ j + n1 , each G is a subgroup of Gj,n1 .

(g) For I + C ∈ Gj,n1 , unique matrices I + V ∈ G exist with

I + C = (I + Vj+n1 ) · . . . · (I + Vj+1 ).

9.2 The Periodicity Relation

The result we are going to prove next says that, of the in¬nitely many

Stokes multipliers, it su¬ces to compute only j0 consecutive ones, for j0 as

de¬ned in Section 8.5. This will be a consequence of the following lemma

showing how the multipliers change when we choose another HLFFS “ in

particular, another fundamental solution Y (z) of the transformed system.

ˆ

Since we have seen that a permutation of the data pairs of an HLFFS F (z)

re¬‚ects in a permutation of its columns, we may without loss of generality

restrict to the case of two HLFFS with the same data pairs.

Lemma 12 Given a system (3.1) with an essentially irregular singular-

ˆ ˆ

ity at in¬nity, let (F (z; 1), Y (z; 1)), (F (z; 2), Y (z; 2)) be two HLFFS with

identical data pairs. Then there is a unique constant invertible matrix

D = diag [D1 , . . . , Dµ ], diagonally blocked in the block structure determined

by the data pairs, for which Y (z; 1) = T (z) Y (z; 2) D, with a q-meromorphic

ˆ

transformation T (z) that is likewise blocked and satis¬es F (z; 1) T (z) =

(1) (2)

ˆ

F (z; 2). For the corresponding Stokes multipliers Vj , Vj we then have

(1) (2)

= D’1 Vj D, for every j ∈ Z.

Vj

Proof: From the Main Theorem (p. 129) we conclude that the matrix

T (z) = F ’1 (z; 1) F (z; 2) is diagonally blocked and convergent. If Fj (z; n)

ˆ ˆ

ˆ

denote the HLNS corresponding to F (z; n), then Fj (z; 1) T (z) = Fj (z; 2)

follows from the properties of k-summability, for every j ∈ Z. Since the

9.2 The Periodicity Relation 143

Fj (z; n) Y (z; n) are fundamental solutions of (3.1), for every j and n, we

conclude Fj (z; 1) Y (z; 1) = Fj (z; 2) Y (z; 2) Cj , for constant invertible Cj .

This implies Y (z; 1) = T (z) Y (z; 2) Cj ; hence Cj is diagonally blocked and

independent of j. Writing D instead of Cj , one may complete the proof. 2

ˆ

Now consider an HLFFS (F (z), Y (z)). Since its data pairs (qj (z), sj ) are

closed with respect to continuation, there exists a permutation matrix R,1

for which Q(z) = diag [q1 (z) Is1 , . . . , qµ (z) Isµ ] = R Q(ze2πi ) R’1 . In fact,

R can be made unique by requiring that, when blocked of type (s1 , . . . , sµ ),

each block of R is either the zero or the identity matrix of appropriate size.

With this block permutation matrix R we now show:

Lemma 13 Let a system (3.1) have an essentially irregular singularity at

ˆ

in¬nity, and let (F (z), Y (z)) be an HLFFS. Then there is a unique con-

stant diagonally blocked matrix D and an equally blocked q-meromorphic

transformation T (z) for which Y (ze2πi ) = T (z) R’1 Y (z) R D, F (ze2πi ) =

ˆ

F (z) R T ’1 (z), and Fj+j0 (ze2πi ) = Fj (z) R T ’1 (z), for j ∈ Z.

ˆ

ˆ

Proof: Observe that (F (z exp[2πi]), Y (z exp[2πi])) is again an HLFFS that

has the same data pairs as (F (z) R, R’1 Y (z) R), and apply Lemma 12 plus

ˆ

2

the properties of k-summability.

We denote R D by exp[2πiL], and refer to it, resp. to L, as the formal

ˆ

monodromy factor, resp. matrix of the HLFFS (F (z), Y (z)).

Proposition 21 Let a system (3.1) have an essentially irregular singular-

ˆ

ity at in¬nity, and let (F (z), Y (z)) be an HLFFS with corresponding Stokes™

multipliers Vj , HLNS Fj (z) and fundamental solutions Xj (z) = Fj (z) Y (z)

of (3.1). Then we have for every j ∈ Z:

Xj+j0 (ze2πi ) = Xj (z) e2πiL , Vj+j0 = e’2πiL Vj e2πiL . (9.4)

Moreover, a monodromy matrix Mj for Xj (z) is given by the identity

e2πiMj = e2πiL (I + Vj+j0 ) · . . . · (I + Vj+1 ). (9.5)

Proof: From Lemma 13 we obtain (9.4). This in turn implies (9.5), repeat-

2

edly using X ’1 (z) = X (z) (I + V ).

Note that (9.5) shows that the computation of a monodromy matrix of

(3.1) is essentially achieved once we have computed the Stokes multipli-

ers. In some cases both problems even are equivalent, as follows from the

exercises below.

1 This is a matrix di¬ering from I by a permutation of its columns, or equivalently,

rows.

144 9. Stokes™ Phenomenon

Exercises: In the following exercises, make the same assumptions as in

the above proposition. In particular, let matrices always be blocked of type

(s1 , . . . , sµ ).

1. Prove that if a matrix C can be factored as C = D C+ C’ , with

D diagonally blocked, and C+ upper, resp. C’ lower triangularly

blocked, both factors having identity matrices along the diagonal,

then the factors are uniquely determined.

2. Enumerate the pairs (n, m), 1 ¤ m < n ¤ µ in any order, and let

all blocks of C on and above the diagonal vanish. Prove I + C =

(I + C1 ) · . . . · (I + Cµ(µ’1)/2 ), with unique matrices Cj whose blocks

all vanish except for that one in position (nj , mj ).

3. Show the following for k = 1: If exp[2πiL] and any one of the matri-

ces exp[2πiMj ] are known, then all the Stokes multipliers Vj can be

computed from the identities established above.

9.3 The Associated Functions

Only in a few cases in dimension ν = 2 shall we succeed in expressing

the Stokes multipliers in terms of known higher transcendental functions

of the parameters of the system; in general their nature seems to be of the

same degree of transcendence as that of the solutions itself. Nonetheless,

it is of importance to learn as much as possible about how to, at least

theroretically, compute them in terms of the data in the system. Here,

we shall study in which sense the Stokes multipliers can be obtained via

an analysis of the singularities of certain associated functions, related to

the HLNS by Borel transform. This is a case of what Ecalle has named

resurgent analysis. While Braaksma [69] and Immink [137] have obtained

equivalent results working with the classical type of FFS, we shall here

follow a presentation in [39], which was based on HLFFS, restricted to

k = 1. For similar results in case of k = r > 1, but a leading term with

distinct eigenvalues, see Reuter [231, 232].

Consider a ¬xed system (3.1) having an essentially irregular singularity at

ˆ

in¬nity, and a likewise ¬xed HLFFS (F (z), Y (z)). To simplify notation, we

shall without loss of generality make the following additional assumptions:

• Assume that k is an integer. This can be brought about by a change

ˆ

of variable z = wq and has the e¬ect that F (z) does not contain

any roots. By such a change of variable, the Stokes directions „j are

replaced by „j /q, but the Stokes multipliers Vj remain the same. For

integer k, the block permutation matrix R introduced above equals

I; hence the formal monodromy matrix L is diagonally blocked, say,

L = diag [L1 , . . . , Lµ ].

9.3 The Associated Functions 145

• Assume that the polynomials qm (z) in the data pairs of the HLFFS

(p)

have the form qm (z) = ’um z k , ’um = »m /k. This can always be

made to hold by a scalar exponential shift. Under this assumption, the

transformed system in the de¬nition of HLFFS on p. 56 has Poincar´ e

rank k. Moreover, the diagonal blocks of Y (z) are of the form Ym (z) =

Gm (z) exp[’um z k ], with G± (z) of exponential growth strictly less

m

than k in arbitrary sectors.

ˆ

• Assume that F (z) is a formal power series with constant term equal

to I. According to Exercise 4 on p. 41, this can be made to hold

by means of a terminating meromorphic transformation T (z), which

does not change the Stokes multipliers, but may lower the Poincar´e

rank of the system. Under this assumption we then have that k = r,

the Poincar´ rank of the system (3.1), and this implies p = 0.

e

A system (3.1), satisfying all the above assumptions, will from now on be

called normalized, and we use the same adjective for an HLFFS of the above

form. For such a normalized HLFFS, we shall de¬ne associated functions.

These functions will have branch points at all the points um , 1 ¤ m ¤ µ;

hence the corresponding Riemann surface will be somewhat di¬cult to

visualize. Therefore, we shall instead consider a complex u-plane together

with cuts from each un to in¬nity along the rays arg(u ’ un ) = ’r d with

arbitrarily ¬xed d. For simplicity, we choose d so that the cuts are all

(p)

disjoint. According to the de¬nition of dj , and observing un = ’»n , this

ˆ

is so if and only if d is a nonsingular direction for F (z) in the sense of

Section 6.4. The set of u not on any one of the cuts shall be denoted by

C d . It is worthwhile to mention that, unlike on a Riemann surface, points

u here are the same once their arguments di¬er by a multiple of 2π.

To de¬ne the associated functions, we ¬rst introduce for every m, with

1 ¤ m ¤ µ:

r r

(u’um )

z s’1 Gm (z) ez

Hm (u; s; k) = dz, (9.6)

2πi β(„ )

for „ = ’d + (2k + 1) π/r with k ∈ Z, and a path of integration β(„ )

consisting of the ray arg z = ’„ ’ (µ + π)/(2r) from in¬nity to the point

z0 with |z0 | = ρ + µ, then on the circle |z| = ρ + µ to the ray arg z =

’„ + (µ + π)/(2r), and back to in¬nity along this ray, for small µ > 0.

Note that the integral is independent of µ, and absolutely and compactly

convergent for s ∈ C and | arg(u ’ um ) ’ r „ | < µ/2. After changes of

variable z ’ 1/z and u ’ ur , the integral (9.6) is nothing but the Borel

transform2 of z s Ym (z).

2 Observethat here it is important to take the original form of the Borel operator, as

de¬ned on p. 80.

146 9. Stokes™ Phenomenon

Because of the choice of „ , the sector of convergence of (9.6) is a subset

of C d . Turning the path of integration by an angle of ±, i.e., replacing

„ by „ ’ ±, obviously results in continuation of Hm (u; s; k) in the oppo-

site sense by the angle ’r±. In particular, the points un for n = m do

not cause any problems as regards continuation of the function! Therefore,

each branch Hm (u; s; k) becomes a holomorphic function in C d , and is

even holomorphic at every point u on the cuts arg(u ’ un ) = ’r d, includ-

ing the point un itself, for every n = m. As a convenient way of stating

this, we shall write Hm (u; s; k) = hol(u ’ un ) for n = m, 1 ¤ n, m ¤ µ.

Moreover, the various branches Hm (u; s; k) are interrelated as follows: The

values of Hm (u; s; k) on the right-hand side of the mth cut3 equal those of

Hm (u; s; k + 1) on the left-hand side. So in other words, holomorphic con-

tinuation of Hm (u; s; k) across the cut in the positive direction produces the

branch Hm (u; s; k + 1). Moreover, from Y (z e2πi ) = Y (z) exp[2πi Lm ] we

conclude via a corresponding change of variable in (9.6) that Hm (u; s; k +

r) = Hm (u; s; k) e’2πi (sI+Lm ) . This shows that r consecutive branches gen-

erate the others via explicit linear relations.

∞

Block the formal series F (z) = 0 Fn z ’n into column blocks Fm (z) =

ˆ ˆ

∞ ’n ˆ

0 Fn,m z of sm consecutive columns of F (z). The series

∞

Fn,m Hm (u; s ’ n; k),

¦m (u; s; k) = (9.7)

n=0

ˆ

according to Exercise 2 and the Gevrey order of F (z), converges absolutely

for s ∈ C and 0 < |u ’ um | < δ, with su¬ciently small δ > 0, and

convergence is locally uniform in u. Following the terminology introduced

in [39], we shall name ¦m (u; s; k) the associated functions to the HLFFS

ˆ

(F (z), Y (z)). In a way, the series (9.7) is the formal Borel transform of

ˆ

index r of Fm (z) Ym (z). As we shall show in Lemma 14, this analogy is

more than formal, since ¦m (u; s; k) will turn out to be the Borel transform

of the corresponding column blocks of the HLNS Xj (z).

Note that (9.7) in principle allows to compute the associated functions

ˆ

¦m (u; s; k) in terms of the HLFFS (F (z), Y (z)) without knowledge of the

HLNS. Thus, we take the attitude of the matrices ¦m (u; s; k) as being

known, at least near the point um . From the above monodromy behavior of

Hm (u; s; k) we immediately obtain that ¦m (u; s; k + 1) is the holomorphic

continuation of ¦m (u; s; k) across the mth cut, in the positive sense, while

¦m (u; s; k + r) = ¦m (u; s; k) e’2πi (sI+Lm ) . (9.8)

To discuss holomorphic continuation of these functions into C d , we ¬rst

establish an integral representation:

3 To understand this or any statement like “a point is to the right of the mth cut,”

rotate Cd about the origin so that the cuts point “upwards,” in which case the meaning

of “left” and “right” is intuitive.

9.3 The Associated Functions 147

For „ = ’d + (2k + 1) π/r with k ∈ Z, there is a unique j = j(k) ∈ Z