8. For c > 0, let f (z) = g(z + π/2 + c). For every ¬xed φ ∈ R, show that

f (reiφ ) remains bounded as r ’ ∞. Why does this not imply that f

is of exponential order zero in every sector S?

4.3 Formal Power Series

Given a sequence (fn )∞ of elements of the Banach space E , the series

n=0

ˆ(z) = ∞ fn z n is called a formal power series (in z), the term “formal”

f 0

emphasizing that we do not restrict the coe¬cients fn in any way; thus the

radius of convergence of the series may well be equal to zero. The set of

ˆ

all such formal power series is denoted by E [[z]]. We say that f converges,

or is convergent, if ρ > 0 exists so that the power series converges for all

z with |z| < ρ, de¬ning a function f (z), holomorphic in D(0, ρ). We shall

ˆ ˆ ˆ

call f (z) the sum of f whenever f converges, and we write f = S f . The

set of all convergent power series will be denoted by E {z}.

ˆ fn z n is a formal power series so that for some positive c, K,

If f (z) =

and s ≥ 0 we have

fn ¤ c K n “(1 + sn) (4.3)

ˆ

for every n ≥ 0, then we say that f is a formal power series of Gevrey

order s, and we write E [[z]]s for the set of all such formal power series.

Compare this to the de¬nition of formal transformations of Gevrey order s

on p. 39. Obviously, (4.3) holds for s = 0 if and only if the power series

converges, hence E [[z]]0 = E {z}. It follows from the exercises below that

E [[z]]s , under natural operations, is a vector space closed under termwise

di¬erentiation, and a di¬erential algebra in case of E being a Banach alge-

bra.

Exercises: For formal power series in E [[z]] we consider the usual oper-

ations; also compare Section B.2 of the Appendix.

1. Show that E [[z]], with respect to addition and multiplication with

scalars, is a vector space over C .

2. In case E is a Banach algebra, show that E [[z]], with respect to mul-

tiplication of power series, is an algebra over C , and is commutative

if and only if E is so.

3. In case E is a Banach algebra, show that E [[z]], with respect to

termwise derivation, is a di¬erential algebra over C ; i.e., show that the

ˆ ˆ ˆˆ ˆ ˆ ˆˆ

map f ’ f is C -linear and obeys the product rule (f g ) = f g + f g .

4. Suppose that E is a Banach algebra with unit element e (hence e = 1

if E = C ). Show that then e, the power series whose coe¬cients are

ˆ

4.4 Asymptotic Expansions 65

all zero except for the constant term being equal to e, is the unit

element in E [[z]].

5. Suppose that E is a Banach algebra with unit element e. Show that

ˆ

the invertible elements of E [[z]], i.e., those f to which g exists such

ˆ

ˆˆ

that f g = e, are exactly those whose constant term is invertible in

ˆ

E (i.e., is nonzero for E = C ).

6. Assume that E is a Banach algebra. For arbitrary s ≥ 0, show that

E [[z]]s , with respect to the same operations as above, again is a dif-

ferential algebra over C .

7. Suppose that E is a Banach algebra with unit element e. For arbitrary

ˆ

s ≥ 0, show f ∈ E [[z]]s invertible (in E [[z]]s ) if and only if it is

invertible in E [[z]], i.e., if and only if its constant term is invertible.

ˆ

8. For arbitrary s ≥ 0, show that for f = fn z n ∈ E [[z]]s with f0 = 0

∞

we have z ’1 f (z) = 0 fn+1 z n ∈ E [[z]]s .

ˆ

4.4 Asymptotic Expansions

ˆ

Given a function f ∈ H(G, E ) and a formal power series f (z) = fn z n ∈

ˆ

E [[z]], one says that f (z) asymptotically equals f (z), as z ’ 0 in G, or:

ˆ

f (z) is the asymptotic expansion of f (z) in G, if to every N ∈ N and

¯ ¯

every closed subsector S of G there exists c = c(N, S) > 0 such that

N ’1

|z|’N f (z) ’ 0 ¯

fn z n ¤ c for z ∈ S. This is the same as saying that

the remainders

N ’1

’N

f (z) ’ fn z n

rf (z, N ) = z

n=0

are bounded at the origin, for every N ≥ 0. If this is so, we write for short

f (z) ∼ f (z) in G, and whenever we do, it will go without saying that G is

=ˆ

ˆ

a sectorial region, f ∈ H(G, E ) and f ∈ E [[z]].

For the classical type of asymptotics with E = C the results to follow

are presented in standard texts as, e.g., [82, 281]. The proofs given there

generalize to the Banach space situation, as we show now:

Proposition 7 Given a sectorial region G, let f be holomorphic in G, with

values in E and so that f (z) ∼ f (z) in G for some f (z) = fn z n ∈ E[[z]].

=ˆ ˆ

Then the following holds true:

(a) The remainders rf (z, N ) are all continuous at the origin, and

rf (z, N ) ’’ fN , z ’ 0, N ≥ 0.

G (4.4)

66 4. Asymptotic Power Series

(b) Suppose that the opening of G is larger than 2π and that f (z) is single-

ˆ

valued. Then f (z) is holomorphic at the origin, and f converges and

coincides with the power series expansion of f (z) at the origin.

Proof: (a) Observe z rf (z, N + 1) = rf (z, N ) ’ fN ; hence rf (z, N + 1)

bounded at the origin implies (4.4).

(b) Under our assumptions, f (z) is a single-valued holomorphic function

in a punctured disc around the origin and remains bounded as z ’ 0. Hence

the origin is a removable singularity of f , i.e., f (z) can be expanded into its

power series about the origin. It follows right from the de¬nition that the

power series expansion is, at the same time, an asymptotic expansion, and

from (a) we conclude that an asymptotic expansion is uniquely determined

ˆ

by f (z). This proves f (z) to converge and be the power series expansion

2

for f (z).

The following result shows that existence of an asymptotic expansion is

equivalent to f ™s being in¬nitely often di¬erentiable at the origin. However,

observe that, unlike the case of holomorphic functions, existence of some

derivative at the origin does not imply existence of higher derivatives.

Proposition 8 Let f be holomorphic in a sectorial region G. Then the

following statements are equivalent:

(a) f (z) ∼ f (z) =

∞

=ˆ fn z n in G.

0

(b) The function f is in¬nitely often di¬erentiable at the origin, and

f (n) (0) = n! fn for n ≥ 0.

(c) All derivatives f (n) (z) are continuous at the origin, and

f (n) (z) ’’ n! fn , z ’ 0, n ≥ 0.

G

Proof: For the equivalence of (b) and (c), compare Exercises 1 and 2 on

p. 63. Assuming (a), observe that according to Cauchy™s integral formula

for derivatives

n! rf (w, n + 1)

dw = f (n) (z) ’ n! fn ,

(1 ’ z/w) n+1

2πi

¯

integrating along a circle around z. For z in a closed subsector S of G, the

radius of this circle can be taken to be µ|z|, for some ¬xed µ, 0 < µ < 1,

¯

depending on S but independent of z. Thus, w/z = 1 + µeiφ for some real

number φ, so that the denominator of the integral can be bounded from

¯

below by some positive constant independent of z, but depending on S.

The standard type of estimate implies that the right-hand side tends to

4.4 Asymptotic Expansions 67

¯

zero uniformly in S; hence (c) follows. Conversely, let (c) hold. Then for

z, z0 ∈ G and N ≥ 0 we have

N ’1

f (n) (z0 ) z

1

(z ’ w)N ’1 f (N ) (w) dw.

f (z) ’ (z ’ z0 )n =

(N ’ 1)!

n! z0

0

Letting z0 ’ 0 and estimating the right-hand side of the resulting formula

2

then implies (a).

Let A(G, E ) be the set of all functions f ∈ H(G, E ), having an asymp-

ˆ

totic expansion f (z). In view of the above proposition, part (a), to every

f (z) ∈ A(G, E ) there is precisely one f ∈ E [[z]] such that f (z) ∼ f (z) in

ˆ =ˆ

G. Therefore, we have a mapping

ˆ

J : A(G, E ) ’’ E [[z]], f ’’ f = J f, (4.5)

mapping each f (z) to its asymptotic expansion. The results to follow show

that A(G, E ), under the natural operations and assuming E a Banach alge-

bra, is a di¬erential algebra, and J is a surjective homomorphism between

A(G, E ) and E [[z]]. However, examples given in the exercises below show

that J is not injective “ even if we consider regions of large opening. In the

next section we are going to study another type of asymptotic expansions

better suited to our purposes, since it will turn out that the corresponding

map J, for regions of su¬ciently large opening, is injective, however, not

surjective.

Theorem 13 Given a sectorial region G, suppose that f1 , f2 ∈ A(G, E ).

Then f1 + f2 ∈ A(G, E ) and J(f1 + f2 ) = J f1 + J f2 . In other words,

fj (z) ∼ fj (z) in G,

=ˆ 1 ¤ j ¤ 2,

implies

f1 (z) + f2 (z) ∼ f1 (z) + f2 (z) in G.

=ˆ ˆ

2

Proof: Follows directly from the de¬nition.

If E is a Banach algebra, then a product of any two elements of A(G, E )

again belongs to A(G, E ). This will be a corollary of the following more

general result:

Theorem 14 Suppose that E , F are both Banach spaces and G is a sec-

torial region. Let f ∈ A(G, E ), ± ∈ A(G, C ), and T ∈ A(G, L(E , F)).

Then

T f ∈ A(G, F), J(T f ) = (J T )(J f ),

± f ∈ A(G, E ), J(± f ) = (J ±)(J f ).

68 4. Asymptotic Power Series

Proof: We only prove the ¬rst one of the two statements; the second one

can be shown following the same steps.

∞ ∞

Let (J T )(z) = 0 Tn z n , (J f )(z) = 0 fn z n . Given a closed subsec-

¯ ¯

tor S of G and N ∈ N, there exists cN > 0 such that for every z ∈ S we

have rf (z, N ) ¤ cN , rT (z, N ) ¤ cN . Then

N ’1

rT f (z, N ) ¤ T (z) rT (z, N ’ m) .

rf (z, N ) + fm

m=0

Proposition 7 (p. 65) part (a) implies fm ¤ cm (m ≥ 0). Hence

N

rT f (z, N ) ¤ cm cN ’m , (4.6)

m=0

2

completing the proof.

If E is a Banach algebra, every f ∈ A(G, E ) can be identi¬ed with an

element of L(E , E ). Thus, the above theorem immediately implies:

Corollary to Theorem 14 In case E is a Banach algebra, let f1 , f2 ∈

A(G, E ). Then f1 f2 ∈ A(G, E ) and J(f1 f2 ) = (J f1 )(J f2 ). In other

words,

fj (z) ∼ fj (z) in G, 1 ¤ j ¤ 2,

=ˆ

implies

f1 (z) f2 (z) ∼ f1 (z) f2 (z) in G.

=ˆ ˆ

Note that f (z) ∼ f (z) implies continuity of f (z) at the origin, so that

=ˆ

z zˆ

ˆ ˆ

f (w) dw is well de¬ned. For f (z) ∈ E [[z]] we de¬ne 0 f (w) dw by

0

termwise integration.