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for which dj’1 < ’„ < dj . By de¬nition of the number n1 in Exercise 4
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 .

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