on p. 142, we have j(k + 1) = j(k) ’ 2 n1 for every k. With this value j(k),

consider the integrals

r r

(u’um )

1 ¤ m ¤ µ,

z s’1 Fj(k),m (z) Gm (z) ez dz, (9.9)

2πi β(„ )

with β(„ ) as in (9.6). Note that for su¬ciently small µ > 0, the path β(„ ) ¬ts

into the sector Sj(k) where Fj(k),m (z) is bounded. So the integral converges

for s ∈ C and | arg(u ’ um ) ’ r „ | < µ/2 and is holomorphic in both

variables. By variation of „ , we can continue the function with respect to

u into the region Gk,m = {z : ’r dj(k) < arg(u ’ um ) < ’r dj(k)’1 } ‚ C d .

As we shall show now, this integral represents ¦m (u; s; k):

Lemma 14 Under the assumptions made above, the integral (9.9) equals

¦m (u; s; k), for arbitrary s ∈ C and u ∈ Gk,m with |u ’ um | su¬ciently

small. Hence ¦m (u; s) is holomorphic in Gk,m .

Proof: Fix a value for „ and restrict to such u with arg(u ’ um ) = r „ .

For N ∈ N, let ¦m (u; s; k; N ) stand for the di¬erence between the integral

and the ¬rst N terms of the right-hand side of (9.7). Observing (9.6),

express ¦m (u; s; k; N ) as a corresponding integral. By estimates completely

analogous to those in the proof of Theorem 23 (p. 80), one can show that

¦m (u; s; k; N ) tends to zero as N ’ ∞, for su¬ciently small values of

|u ’ um |. The identity theorem then completes the proof. 2

To discuss the global behavior of ¦m (u; s; k) in C d , we introduce auxiliary

functions ¦— (u; s; k) as follows: For every k ∈ Z, there exists a unique

m

j = j (k) ∈ Z with dj’1 < d ’ 2k π/r < dj , and we have j — (k) = j(k) + n1 ,

—

with j(k) and n1 as above. With this value j — (k) and any ¬xed point z0 of

modulus larger than ρ and independent of k, we de¬ne

∞(±)

r r

¦— (u; s; k) (u’um )

z s’1 Fj — (k),m (z) Gm (z) ez

= dz,

m

2πi z0

integrating along the circle |z| = |z0 | to the ray arg z = ±, and then along

this ray to in¬nity. For dj — (k)’1 ’π/(2r) < ± < dj — (k) +π/(2r), this integral

converges compactly for s ∈ C and π/2 < r ± + arg(u ’ um ) < 3π/2.

Varying ±, the function can be continued into ’r dj — (k) < arg(u ’ um ) <

2π ’ r dj — (k)’1 . By de¬nition of j — (k), we therefore see that ¦— (u; s; k) is

m

holomorphic in C d , and even holomorphic at points on the nth cut, for

every n = m, including the point un itself. Using the same notation as

above, we write ¦— (u; s; k) = hol(u ’ un ), for n = m, 1 ¤ n, m ¤ µ.

m

Unlike the associated functions, the auxiliary ones are not directly linked

via holomorphic continuation across the mth cut. However, from (9.4) one

148 9. Stokes™ Phenomenon

obtains that

¦— (u; s; k + r) = ¦— (u; s; k) e’2πi(sI+Lm ) , u ∈ C d. (9.10)

m m

The ¦— (u; s; k) are linked to the auxiliary functions as follows:

m

Lemma 15 For every k ∈ Z and every m with 1 ¤ m ¤ µ, the associated

function ¦m (u; s; k) is holomorphic in C d . Moreover, we have

¦m (u; s; k) = ¦— (u; s; k) ’ ¦— (u; s; k + 1) + hol(u ’ um ).

m m

Proof: Let „ and j(k) be as in (9.9). The sectors Gk,m all have paral-

lel bisecting rays, so their intersection Gk is not empty. For u ∈ Gk , let

¦(u; s; k) = [¦1 (u; s; k), . . . , ¦µ (u; s; k)], and de¬ne ¦— (u; s; k) analogously.

From (9.9) we then conclude

∞(±+ ) ∞(±’ )

r r

’ z s’1 Xj(k) (z) ez u

¦(u; s; k) = dz,

2πi z0 z0

with ±± = ’„ ± (π + µ)/(2r), and Xj(k) (z) as in Section 9.1. While the

two paths of integration have the same form as in the de¬nition of the

auxiliary functions, the index j(k) is not correct: There we may only use

values j, di¬ering from j(k) by odd multiples of n1 , and so that the path of

integration ¬ts into Sj . For ±+ , this is correct for j — (k) = j(k) + n1 , while

for ±’ this is so with j — (k+1) = j(k)’n1 . According to the de¬nition of the

+ ’

Stokes multipliers, Xj(k) (z) = Xj — (k) (z) (I + Ck ) = Xj — (k+1) (z) (I + Ck ),

+ ’

with I + Ck = (I + Vj(k)+n1 ) · . . . · (I + Vj(k)+1 ), (I + Ck )’1 = (I + Vj(k) ) ·

. . . · (I + Vj(k)’n1 +1 ). Consequently we have

+ ’

¦(u; s; k) = ¦— (u; s; k) (I + Ck ) ’ ¦— (u; s; k + 1) (I + Ck ). (9.11)

This implies holomorphy of ¦(u; s; k) in C d . From Exercise 4 on p. 142

+ ’

we obtain that both matrices Ck and Ck have vanishing diagonal blocks.

Since ¦— (u; s; k) and ¦— (u; s; k + 1) both are holomorphic at um whenever

n n

2

n = m, the proof is completed.

Formula (9.11) contains more than the information we have used to prove

the lemma: Let us write m ≺ n, whenever the nth cut is located to the

right of the mth one.4 Consider any (n, m) ∈ Suppj(k),n1 , i.e., in view of

Exercise 4 on p. 142, any (n, m) ∈ Supp with j(k) + 1 ¤ ¤ j(k) + n1 .

From the de¬nition on p. 141, we conclude that this holds if and only if

arg(un ’ um ) = ’r d ’1 . By de¬nition of j(k), one ¬nds that the point un ,

when pictured in C d , is located in the sector ’r d’π < arg(u’um ) < ’r d.

C d,

4 As explained earlier, this is to be understood after a rotation of which makes

the cuts point upward.

9.3 The Associated Functions 149

So (n, m) ∈ Suppj(k),n1 holds if and only if n ≺ m. Similarly, one ¬nds

(k)

(n, m) ∈ Suppj(k)’n1 ,n1 if and only if m ≺ n. In both cases, we write Cn,m

+ ’

for the corresponding block of Ck , resp. Ck . Using this terminology, we

state the following central result for the behavior of the associated functions

at the points un :

Theorem 46 Let a normalized system (3.1) with a normalized HLFFS

ˆ

(F (z), Y (z)) be given, and let d be a nonsingular direction. For every n = m

and every s ∈ C for which (I ’ e2πi (sI+Ln ) )’1 exists, the corresponding

associated functions satisfy

r’1

¦n (u; s; k + ) (I ’ e’2πi (sI+Ln ) )’1 Cnm + hol(u ’ un ),

(k)

¦m (u; s; k) =

=0

whenever n ≺ m in the above sense, while in the opposite case we have

r

¦n (u; s; k + ) (e’2πi (sI+Ln ) ’ I)’1 Cnm + hol(u ’ un ).

(k)

¦m (u; s; k) =

=1

Proof: For the proof, we shall restrict ourselves to the ¬rst case; the argu-

ments in the second are very much analogous. Recall that ¦— (u; s; k) is holo-

n

±

morphic everywhere except at un . Using this, together with the form of Ck ,

(k)

we conclude from (9.11) that ¦m (u; s; k) = ¦— (u; s; k) Cnm + hol(u ’ un ).

n

—

r’1

From Lemma 15 and (9.10) we conclude =0 ¦n (u; s; k+ ) = ¦n (u; s; k)’

¦— (u; s; k + r) + hol(u ’ un ) = ¦— (u; s; k) (I ’ e’2πi (sI+Ln ) ) + hol(u ’ un ),

n n

2

completing the proof.

This theorem has the following consequence:

Corollary to Theorem 46 Under the assumptions of the above theorem,

let ¦m,n (u; s; k) stand for the continuation of ¦m (u; s; k) across the nth cut

in the positive, resp. negative, sense whenever n ≺ m, resp. m ≺ n. Then

in both cases

(k)

¦m (u; s; k) ’ ¦m,n (u; s; k) = ¦n (u; s; k) Cnm . (9.12)

Proof: Observe that ¦n (u; s; k + ), when continued across the nth cut in

the positive sense, becomes ¦n (u; s; k + 1 + ), while terms holomorphic at

2

un remain unchanged.

The above identities theoretically may be used to compute all Stokes™

multipliers of highest level. This shall be investigated in some detail in

Section 9.5.

150 9. Stokes™ Phenomenon

Exercises: In what follows, consider a ¬xed normalized system (3.1) and

ˆ

a likewise ¬xed normalized HLFFS (F (z), Y (z)), and observe the notation

(1) (µ)

introduced above. Moreover, let Bn = diag [Bn , . . . , Bn ] be the coe¬-

cients of the transformed system in the de¬nition of HLFFS on p. 56.

1. For ¬xed u ∈ C d , k ∈ Z, and 1 ¤ m ¤ µ, show that Hm (u; s; k)

satis¬es the following di¬erence equation in the variable s ∈ C :

(m)

n0

s Hm (u; s; k) + ru Hm (u; s + r; k) = ’ n=0 Bn Hm (u; s + r ’ n; k).

2. For m, s, and k as above, and arbitrary δ > 0, show existence of

c, K > 0 so that Hm (u; s ’ n; k) ¤ c (K|u ’ um |)n/r /“(1 + n/r) for

every n ∈ N0 and every u ∈ C with |u ’ um | ≥ δ.

9.4 An Inversion Formula

Since the associated functions have been shown to be the Borel transform

of some of the normal solutions, we may ask for a corresponding inversion

formula. However, in general this will not be a normal type of Laplace

integral, owing to the kind of singularity of ¦m (u; s; k) at um . So instead

of a ray, we shall use a path of integration γm (d), very much like the one in

Hankel™s formula: From in¬nity along the left border of the mth cut until

some point close to um , then on a small, positively oriented circle around

um , and back to in¬nity along the right border of the same cut. With this

path, consider the integral

r’1

r

¦m (u; s; k + ) e’z (u’um )

du. (9.13)

γm (d) =0

From the de¬nition of the auxiliary functions we conclude that they are of

exponential growth at most r everywhere in C d , including the boundaries

of the cuts. From (9.11) we obtain that the same is true for the associated

functions. This then shows that the integral converges in a sectorial region

G at in¬nity of opening π/r and bisecting direction d. There we can identify

the integral with one of the normal solutions:

ˆ

Theorem 47 Let a system (3.1) with a normalized HLFFS (F (z), Y (z)) be

given, and let d be a nonsingular direction. Then for k ∈ Z and 1 ¤ m ¤ µ,

the integral (9.13) equals z s’r Fj — (k),m (z) Gm (z) (I ’ e’2πi(sI+Lm ) ) on the

region G.

— ’2πi (sI+Ln )

r’1

=0 ¦n (u; s; k+ ) = ¦n (u; s; k) (I ’e

Proof: Recall )+hol(u’

un ) from the proof of Theorem 46. Then, use Exercise 2 and observe that an

9.5 Computation of the Stokes Multipliers 151

r

integral γm (d) ¦(u) e’z (u’um ) du vanishes whenever ¦(u) is holomorphic

2

along the path as well as inside the circle about um inside of γm (d).

The above inversion formula is interesting in its own right but will mainly

serve in the next section to show that the columns of ¦m (u; s, k) are linearly

independent.

Exercises: In what follows, make the same assumptions as in the above

theorem.

1. For z0 ∈ Sj—(k) , rewrite the integral representation of ¦— (u; s; k) as

m

a Laplace integral of order 1.

r

¦— (u; s; k) e’z (u’um )

2. Show z s’r Fj — (k),m (z) Gm (z) = du.

γm (d) m

9.5 Computation of the Stokes Multipliers

At the end of the previous section we obtained several identities showing

how the behavior of the associated functions at the singularities un can be

described in terms of the Stokes multipliers of (3.1) (p. 37). Here, we wish

to show that this behavior in turn determines all the Stokes multipliers. To

(k)

do this, we ¬rst prove that (9.12) determines the matrices Cnm uniquely,

owing to linear independence of the columns of ¦m (u; s; k).

ˆ

Lemma 16 Let a system (3.1) with a normalized HLFFS (F (z), Y (z)) be

given, and let d be a nonsingular direction. Then for every m = 1, . . . , µ,

k ∈ Z, and s ∈ C so that I ’exp[’2πi(sI +Lm )] is invertible, the following

is correct: If ¦m (u; s; k) c = hol(u ’ um ) for some c ∈ C sm , then c = 0

follows. In particular, the columns of ¦m (u; s; k) are linearly independent

functions of u ∈ C d .