∞ ∞

un’r un’r

A(u) = An , T (u) = Tn ,

“(n/r) “(n/r)

n=1 n=N

we see that (3.4), (3.2) are equivalent to A(u), T (u) having positive ra-

dius of convergence. Slightly violating previous notation, we set B(u) =

N +M n’r

n=1 Bn u /“(n/r) and abbreviate

u

A (ur ’ tr )1/r T (t) ’ T (t) B (ur ’ tr )1/r dtr .

I(u, T, B) =

0

Then the integral equation

T (u) A0 ’ (A0 + rur I) T (u) = A(u) ’ B(u) + I(u, T, B) (3.17)

can, by termwise integration and use of the Beta Integral (B.11) (p. 229), be

seen to be equivalent to (3.16). So the proof of the theorem is equivalent to

showing existence of T (u) and B(u), of the desired form, satisfying (3.17).

For this, we use the space V of pairs (T, B) of matrix-valued functions that

are holomorphic in R(0, ρ), continuous up to the circle |u| = ρ, for ρ > 0

to be selected later, and have a zero resp. a pole at the origin of order at

least N ’ r resp. at most r ’ 1. This is a Banach space with respect to the

norm

(T, B) = sup ( T (u) |u|r’N + B(u) |u|r’1 ).

|u|¤ρ

˜˜

Given any (T, B) ∈ V , we aim at ¬nding a new pair (T , B) ∈ V so that

˜ ˜ ˜

T (u)A0 ’ (A0 + rur I)T (u) = A(u) ’ B(u) + I(u, T, B). (3.18)

˜

Obviously, (3.18) is a system of linear equations for the matrix T (u). How-

ever, its coe¬cient matrix has a determinant that is a polynomial in u

vanishing at the origin; hence, the inverse matrix is rational and has a

pole, say, of order p, at the origin, and for su¬ciently small ρ > 0, the

inverse matrix is holomorphic in R(0, ρ) and continuous along the circu-

˜˜

lar boundary. Hence the pair (T , B) belongs to V , if the right-hand side

of (3.18) vanishes at the origin of order at least N + p ’ r. By choosing

˜

B(u) appropriately, we can arrange the right-hand side to vanish of order

¯

N + M ’ r (and then B(u) is unique); hence we choose M = p. Thus,

54 3. Highest-Level Formal Solutions

we have de¬ned a mapping of V into itself, and a ¬xed point is a solution

of (3.17). To apply Banach™s ¬xed point theorem, we need the following

estimates (for |u| ¤ ρ):

Taking any pair (T, B) with norm at most k, we have by de¬nition of

the norm in V that T (u) ¤ k |u|N ’r , B(u) ¤ k |u|1’r and, for suitable

a ≥ 0, A(u) ¤ a |u|1’r . Integrating along the straight line segment from

0 to u and using the Beta Integral we obtain

“(1/r)“(N/r)

I(u, T, B) ¤ cN (a + k) k |u|N +1’r , cN = .

“((N + 1)/r)

N

˜ ˜

Putting B(u) = n=1 An un’r /“(n/r) + B2 (u), we ¬nd, using Cauchy™s

integral formula, that

1 dt

˜

B2 (u) = [A2 (t) + I(t, T, B)]k(u/t) ,

2πi t

|t|=ρ

∞ n’r

with A2 (t) = n=N +1 An u /“(n/r) and the kernel function k(w) =

wN +1’r (1 ’ wM )/(1 ’ w). Since M = sup|w|¤1 |1 ’ wM |/|1 ’ w|, we can es-

timate the above integral to obtain B2 (u) ¤ M [a2 +cN (a+k) k]|u|N +1’r ,

˜

with a2 depending only on A2 (u). Hence the right-hand side of (3.18) is at

most (M + 1)(a2 + cN (a + k) k)|u|N +1’r (but note that by choice of B2 (u),

˜

the right-hand side vanishes at the origin of order N + M ’ r). For u on

the circle |u| = ρ, we have

˜

|u|r’N T (u) ¤ c(a2 + cN (a + k) k)

with a constant c that depends on ρ and M , but is independent of N . Due

to the maximum principle, the same estimate then holds for smaller |u|.

Combining the above estimates, one then obtains

˜˜

(T , B) ¤ b1 + b2 cN (a + k)k,

with constants bj that depend on A(u), but are independent of N . Since

cN ’ 0 as N ’ ∞, one ¬nds that for k = 2b1 we may choose N so large

˜˜

that the norm of (T , B) is at most equal to k. In other words, the mapping

˜˜

(T, B) ’’ (T , B) maps the ball of V of radius k into itself. By similar

estimates one can show that the mapping is contractive and hence has a

2

unique ¬xed point.

The problem of computing the transformation in the previous theorem

is of a di¬erent nature than for those transformations in previous sections,

since there it was, at least theoretically, possible to give the exact values

of any ¬nite number of coe¬cients, while here we use Banach™s ¬xed point

theorem, which is constructive, but gives approximate values only. However,

since the earlier coe¬cients of our system remain unchanged, this will be

of little in¬‚uence on what we have in mind to do. Therefore, we may say

3.5 Highest-Level Formal Solutions 55

that the previous theorem and the other results of this section reduce the

problem of ¬nding a formal fundamental solution for a system (3.1) to the

same problem for a ¬nite number of systems of the same kind, but of strictly

smaller dimension. In the next section we brie¬‚y indicate a modi¬cation of

the de¬nition of the term formal fundamental solution that was ¬rst given

in a di¬erent, but equivalent, formulation in [8] and which is very natural to

consider, in particular in the discussion of Stokes™ phenomena in sections to

follow. Since Stokes™ multipliers also can at best be approximated, it does

not hurt in this context to have only approximate values for the coe¬cients

of the system.

Exercises: In the following exercises, let a system (3.3) of Gevrey order

s = 1/r be given.

1. Assume that all coe¬cients An of (3.3) commute with one another.

Prove that then Theorem 11 (p. 52) holds, with N = 1 and M = r ’1

ˆ ˆ

and a transformation T (z) commuting both with A(z) and B(z). Note

that this in particular settles the case of dimension ν = 1.

2. Assuming that A0 is diagonal, ¬nd the pole order p of the inverse

matrix corresponding to the left hand side of (3.18).

3.5 Highest-Level Formal Solutions

The classical notion of regular versus irregular singularities distinguishes

between cases where fundamental solutions have only a moderate growth

rate, resp. grow exponentially. As shall become clear later on, the second

case can naturally be divided into two subcases: Sometimes, the exponen-

tial growth is entirely due to the presence of one scalar exponential poly-

nomial in a formal fundamental solution, while in other cases several such

polynomials occur. Therefore, we shall use the following terminology:

1. We say that in¬nity is an almost regular-singular point of (3.1) if

one can ¬nd a terminating meromorphic transformation T (z) so that

∞

the transformed system has the form B(z) = z r n=0 Bn z ’n , with

Bj = »j I for 0 ¤ j ¤ r ’ 1. Note that then a scalar exponential shift

transforms this system into one having a singularity of ¬rst kind at

in¬nity.

2. We say that in¬nity is an essentially irregular singularity of (3.1) if

it is not almost regular-singular.

ˆ

3. A formal q-meromorphic transformation F (z), q ∈ N, is called a for-

mal fundamental solution of highest level for (3.1), if the following

holds:

56 3. Highest-Level Formal Solutions

(a) The coe¬cient matrix B(z) = F ’1 (z)[A(z)F (z) ’ z F (z)] of the

ˆ ˆ ˆ

transformed system is of the form

n0

Bn w’n ,

qr

z = wq ,

B(z) = w

n=0

for some n0 ∈ N, and the coe¬cients Bn are all diagonally

blocked of some type (s1 , . . . sµ ), independent of n, with µ ≥ 2.

(b) For some integer p, 0 ¤ p < rq, which in case q ≥ 2 is positive

and co-prime with q, the coe¬cients B0 , . . . , Bp have the form

(0 ¤ j ¤ p ’ 1),

Bj = »j I

(p)

= diag [»1 Is1 + N1 , . . . , »(p) Isµ + Nµ ],

Bp (3.19)

µ

(p)

with distinct complex numbers » and nilpotent matrices N .

ˆ

(c) The transformation F (z) is of Gevrey order s = q/(qr ’ p).

In the sequel, we shall abbreviate the term highest-level formal fundamen-

tal solution by HLFFS. The notion of HLFFS will turn out to be very

important in what follows, so we wish to make the following comments:

• Having computed an HLFFS for a system (3.1), we have (formally)

partially decoupled the system in the sense that we have reduced the

problem of computing a fundamental solution of (3.1) to the same

problem for several smaller systems. As we shall show later, the di-

ˆ

vergence of the transformation F (z) does not keep us from giving a

clear analytic meaning to it, so that the process of decoupling will be

not only a formal one.

• In the de¬nition of HLFFS, the parameters q, p, »0 , . . . , »p’1 and the

(p)

pairs (»j , sj ), 1 ¤ j ¤ µ, occurred. As a convenient way of referring

to those quantities, we de¬ne

p’1

z r’n/q r’p/q

(p) z

1 ¤ j ¤ µ.

qj (z) = »n + »j ,

r ’ n/q r ’ p/q

n=0

These are polynomials in the qth root of z that shall play an impor-

tant role later on, and the pairs (qj (z), sj ) will be referred to as data

pairs of the HLFFS. We shall show in Chapter 8 that any two HLFFS

of the same system (3.1) have the same data pairs up to a renumer-

ation; in fact, more cannot hold in general, since given one HLFFS,

we can block its columns into blocks of sizes sk , and permuting these

blocks can be seen to produce another HLFFS with the data pairs

permuted correspondingly.

3.5 Highest-Level Formal Solutions 57

• We shall say that the data pairs of any HLFFS are closed with re-

spect to continuation, provided that to every pair (qj (z), sj ), there

is a (unique) pair (q˜ (z), s˜ ) with q˜ (z) = qj (ze2πi ) and s˜ = sj .

j j j j

Obviously this is equivalent to saying that »j = 0 whenever j is

(p)

not a multiple of q, and that the pairs (»j , sj ) are closed with

respect to multiplication by exp[2πi/q], i.e., for suitable ˜ we have

j

(p) (p)

(»˜ , s˜ ) = (»j exp[2πi/q], sj ). We shall show in Chapter 8 that

j

j

this always holds. Also note that, owing to the fact that p, q are co-

prime when p = 0, we can recover p, q from the data pairs of an

HLFFS.

Assuming that in¬nity is an essentially irregular singularity implies that in

the algorithm described in Section 3.3 we end with case 1. This leads to

the following result:

Theorem 12 Every system (3.1) having an essentially irregular singular-

ity at in¬nity possesses an HLFFS whose data pairs are closed with respect

to continuation.

Proof: From the algorithm formulated in Section 3.3, followed by an ap-

plication of the Splitting Lemma together with Theorem 11 (p. 52), we

ˆ

conclude existence of a transformation x = eq(z) T (z) x, with a polynomial

˜

ˆ

q(z) of degree at most r and a formal q-meromorphic transformation T (z)

of Gevrey order s = q/(qr ’ p), p ≥ 0 and co-prime with q provided p = 0,

such that the transformed system, after the change of variable z = wq ,

˜

has a coe¬cient matrix of the form B(w) = wqr’p 0 0 Bn z ’n , with lead-

n˜

˜

˜

ing term B0 having the form required for Bp . The scalar exponential shift

q

exp[’q(w )] then leads to a system satisfying all the requirements for con-

ˆ

cluding that T (z) is an HLFFS, and its data pairs are closed with respect

2

to continuation.

The results of the previous sections enable us to compute an HLFFS in

the following sense:

• First, follow the steps in the algorithm on p. 50 to compute a system

whose leading term has several eigenvalues.

• Next, apply the Splitting Lemma to obtain several formal systems of

Gevrey order 1/˜, with r being their Poincar´ rank.

r ˜ e

• Finally, apply the iteration outlined in the proof of Theorem 11 (p. 52)