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