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for n ≥ N , setting Tn = 0 for n < N . Using the notation
∞ ∞
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-

˜
˜
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)

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