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of Gevrey order s.

6. A transformation T (z) will be called q-analytic, resp. q-meromorph-
ic, with q ∈ N, if T (z q ) is an analytic, resp. meromorphic, trans-

formation, or in other words, if T (z) = n=’n0 Tn z ’n/q , |z| > ρ,
and det T (z) is not the zero series. Analogously, we de¬ne formal q-
ˆ
analytic resp. q-meromorphic transformations T (z). Finally, we say
ˆ ˆ
that T (z) is formal of Gevrey order s if T (z q ) is formal of Gevrey
order s/q, i.e., if the coe¬cients Tn satisfy (3.2) with s/q in place of
s. To see that this is a natural terminology, check what happens to
the Gevrey order of an analytic resp. meromorphic transformation
under a change of variable z = wq .

Note that all the above transformations have inverses of the same type:
For formal analytic transformations of Gevrey order s compare Exercise 3,
and for meromorphic ones use Proposition 5 (p. 40); observe, however, that
the inverse of a terminating transformation will, in general, not terminate.
Proceeding formally, the change of variable x = T (z) x transforms (3.1)
˜
into the transformed system z x = B(z) x, with
˜ ˜

z T (z) = A(z) T (z) ’ T (z) B(z).
40 3. Highest-Level Formal Solutions

However, except for analytic transformations, the transformed system will
in general not be of the same form as (3.1): In case of a formal analytic
transformation the coe¬cient matrix will be given as a series of the form
(3.1), but the radius of convergence of this series will in general be equal to
zero. Systems where this happens, or to be precise: where this may happen,
will be named formal systems and denoted as

An z ’n .
ˆ ˆ
A(z) = z r
z x = A(z) x, (3.3)
n=0

In particular, if for su¬ciently large c, K > 0 we have

An ¤ c K n “(1 + sn), n ≥ 0, (3.4)

then we shall call (3.3) a formal system of Gevrey order s. Applying a
ˆ
formal analytic or meromorphic transformation T (z) to (3.3) and denoting
ˆ
the resulting formal system by z x = B(z) x, the coe¬cient matrices are
˜ ˜
related by the purely formal identity
ˆ ˆ ˆ ˆ ˆ
z T (z) = A(z) T (z) ’ T (z) B(z). (3.5)

Using Exercise 2, we see that a formal analytic transformation of Gevrey
order s transforms a system that is formal of Gevrey order s and of Poincar´
e
rank r into another such system. Shearing transformations, or q-analytic
transformations, take (3.1) or (3.3) into a system that, after a change of
the independent variable as in Exercise 5, is of the same form, but perhaps
of larger rank. Scalar exponential shifts also increase the Poincar´ rank of
e
the system, except when the degree of q(z) is at most r.
For meromorphic transformations, we prove the following well-known
result, showing that they are a combination of analytic ones and a shearing
transformation:
ˆ
Proposition 5 Every formal meromorphic transformation T (z) can be
factored as
ˆ ˆ ˆ
T (z) = T1 (z) diag [z k1 , . . . , z kν ] T2 (z),
ˆ ˆ
where T1 (z), T2 (z) are formal analytic transformations and kj ∈ Z, k1 ¤
ˆ ˆ
k2 ¤ . . . ¤ kν . When T (z) is of Gevrey order s ≥ 0, then both Tj (z) can be
chosen to be of Gevrey order s as well.

Proof: We proceed by induction with respect to ν: For ν = 1, the statement
ˆ ˆ ˆ
obviously holds with T1 (z) ≡ 1 and K = k = deg T (z), where deg T (z)
ˆ
denotes the highest power of z occurring in the series T (z). For ν ≥ 2, let
ˆ ˆ
ˆ ˆ
T (z) = [tjk (z)] and take kν = deg T (z) = maxj,k deg tjk (z). Interchanging
ˆ
rows and columns, we can arrange that kν = deg tνν (z). Adding suitable
multiples of the last row/column to previous ones, the factors used being
3.1 Formal Transformations 41

formal power series in z ’1 , and then dividing the last row by a power series
with nonzero constant term, we can arrange that tνν (z) ≡ z kν , tνj (z) =
ˆ ˆ
tjν (z) ≡ 0, 1 ¤ j ¤ ν ’ 1. All the steps used to obtain this form can be
ˆ
interpreted as multiplication from the left- or right-hand side by formal
ˆ
analytic transformations, which are of Gevrey order s ≥ 0 if T (z) is formal
of Gevrey order s. The induction hypothesis then completes the proof. 2

While the usefulness of formal transformations is not clear o¬hand, the
other types of transformations take (3.1) into an equivalent system in the
sense that it su¬ces to compute a fundamental solution of either one of the
two equations, since then one for the other system is obtained through the
transformation. We shall see in Chapter 8 that the same applies to formal
transformations of Gevrey order s as well, since we shall give a holomorphic
interpretation to the formal series by which formal transformations are
de¬ned. In this chapter, however, we shall take a formal approach, meaning
that we most of the time disregard the question of convergence of formal
power series occurring in calculations, but we will always verify estimates
of the form (3.2) resp. (3.4).

Exercises:
1. Use the Beta Integral (p. 229) to show “(1+x)“(1+y)/“(1+x+y) =
x B(x, 1 + y) ¤ 1 for x, y > 0.

ˆ ˆ
2. If T1 (z), T2 (z) are formal analytic transformations of Gevrey order
ˆ ˆ
s ≥ 0, show the same for T1 (z) T2 (z).

ˆ
3. For a formal analytic transformation T (z) of Gevrey order s, show
that T ’1 (z) is of Gevrey order s, too.
ˆ

4. Show that every formal meromorphic transformation can be factored
ˆ ˆ
as T (z) T (z), with a formal analytic transformation T (z) and a ter-
minating meromorphic transformation T (z). Conclude that in Propo-
ˆ
sition 5 we may take T1 (z) to converge.

5. Show that the system equivalent to (3.3) by means of the change of
variable z = wq , q ∈ N, has Poincar´ rank qr.
e

6. Show that for s > 0 the following statements both are equivalent to
(3.2), possibly for di¬erent values of c, K:

(a) Tn ¤ c K n nsn , n ≥ 1. (b) lim sup Tn /“(1 + sn) < ∞.
n
42 3. Highest-Level Formal Solutions

3.2 The Splitting Lemma
Roughly speaking, ¬nding a formal fundamental solution of (3.1) will be
ˆ
equivalent to ¬nding a formal q-meromorphic transformation T (z) so that
q
the transformed system, after a change of variable z = w , is elementary in
the sense of Section 1.6. A fundamental solution G(z) of the transformed
ˆ ˆ
system can then be easily computed. The object X(z) = T (z) G(z) formally
satis¬es (3.1), and classically these formal fundamental solutions have been
the starting point of the so-called asymptotic theory. In the light of recent
results, it will be more natural to base these investigations on a formal
solution of highest level. This essentially will be a formal q-meromorphic
ˆ
transformation T (z) of a certain minimal Gevrey order, block-diagonalizing
the system (3.1). The diagonal blocks of the transformed system will, in
general, not be elementary. However, since they are of smaller dimensions
than the original system, we shall be able to obtain signi¬cant results simply
by induction with respect to the dimension of the system.
When the leading term A0 of (3.1) has several distinct eigenvalues, exis-
tence of such a transformation, with q = 1, follows from a classical result:

Lemma 3 (Splitting Lemma) Let (3.3) be a formal system of Gevrey
(11) (22)
order s, and assume that A0 = diag [A0 , A0 ], such that the two diagonal
blocks have disjoint spectra. Then there exists a unique formal analytic
transformation of Gevrey order s = max{s, 1/r} of the form
˜

ˆ
I T12 (z)
Tn z ’n ,
(jk)
ˆ ˆ
T (z) = , Tjk (z) =
ˆ
T21 (z) I 1

such that the transformed formal system is diagonally blocked, with each of
the two diagonal blocks being a formal system of Gevrey order s.
˜

Proof: Blocking

ˆ ˆ ˆ
A11 (z) A12 (z) B11 (z) 0
ˆ ˆ
A(z) = , B(z) = ,
ˆ ˆ ˆ
A21 (z) A22 (z) 0 B22 (z)

and inserting into (3.5) leads to

ˆ ˆ ˆ ˆ
B22 (z) = A22 (z) + A21 (z) T12 (z), (3.6)


ˆ ˆ ˆ ˆ ˆ ˆ
z T12 (z) = A12 (z) + A11 (z) T12 (z) ’ T12 (z) A22 (z)
ˆ ˆ ˆ
’T12 (z) A21 (z) T12 (z), (3.7)

plus two other equations with indices 1, 2 permuted that are omitted here
but can be treated in quite the same way. Inserting power series expansions
3.2 The Splitting Lemma 43

and comparing coe¬cients implies
n’1
(12) (22) (11) (12) (11) (22)
(12) (12)
’ (An’m Tm ’ Tm An’m )
T n A0 A0 T n =
m=1
n’µ’1
n’2
(21)
(12) (12)
’ Tµ An’m’µ Tm
µ=1 m=1
(12)
+A(12) + (n ’ r)Tn’r , (3.8)
n

(12)
for n ≥ 1, interpreting Tn’r = 0 for n ¤ r. From these formulas we
(12)
can uniquely compute the coe¬cients Tn . So we are left to estimate the
(12)
coe¬cients Tn , and this can be done as follows (also compare the proof
of Lemma 2 (p. 28)):
It su¬ces to consider the case s ≥ 1/r, so that s = s. By assumption we
˜
(12) ’n
have (3.4). Taking tn = Tn K /“(1 + sn), we conclude from (3.8), for
su¬ciently large c > 0:
˜

n’1
“(1 + s(n ’ m))“(1 + sm)
¤ c 1+2
tn ˜ tm
“(1 + sn)
m=1
n’µ’1
n’1
“(1 + sµ)“(1 + s(n ’ µ ’ m))“(1 + sm)
+ tµ tm
“(1 + sn)
µ=1 m=1
(n ’ r)“(1 + s(n ’ r))
+K ’r n ≥ 1.
tn’r ,
“(1 + sn)
According to Exercise 1 on p. 41, both quotients of Gamma functions inside
the two sums can be estimated by 1, and by taking K larger we may also
assume the term in front of tn’r to be bounded by 1. Finally, by allowing
tn to become larger, we may assume equality in the above estimate, and
n’1 n’µ’1
thus get tn = c [1 + tn’r + µ=1 tµ (2 + m=1 tm )], for n ≥ 1. De¬ning
˜

f (z) = 1 tn z n , we conclude formally
(1 ’ z)f (z) = z c [1 + (2 + z r’1 (1 ’ z)) f (z) + f 2 (z)].
˜
This quadratic equation has exactly one solution that is holomorphic at the

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