¬rst N coe¬cients will be known exactly, since they agree with cor-

responding coe¬cients of the formal systems, while the remaining

nonzero coe¬cients can in principal be computed up to any degree

of accuracy.

58 3. Highest-Level Formal Solutions

ˆ

The HLFFS F (z) obtained by this algorithm is “known” in the sense that

its ¬rst N terms are computed exactly, while ¬nitely many other terms

may be computed up to any desired degree of accuracy. Also, observe that

the above algorithm computes an HLFFS for which the numbers »k in the

de¬nition vanish whenever k is not a multiple of q, so that the data pairs

of the HLFFS are closed with respect to continuation. We shall show that

ˆ

the power series for F (z) in general has radius of convergence equal to

zero, but is k-summable in Ramis™ sense, for k = 1/s. To do so, we shall in

the following chapters provide the necessary instruments from the general

theory of asymptotic power series. In Chapter 8 we then will return to the

investigation of HLFFS.

Exercises:

1. For ν = 1, show that in¬nity always is an almost regular-singular

point of (3.1).

2. Compute an HLFFS for (3.1), with A(z) = zA0 + A1 and

»1 0 ab

(a) A0 = , A1 = , »1 = » 2 .

0 »2 cd

» 0 ab

(b) A0 = , A1 = , b = 0.

1 » cd

3. Assume that we are given a system (3.1) with A0 having all distinct

ˆ

eigenvalues. Verify that the matrix T (z) in the formal fundamental

solution obtained in Exercise 4 on p. 45 is an HLFFS.

4

Asymptotic Power Series

In this chapter we de¬ne the notion of asymptotic expansions, in particular

Gevrey asymptotics, and we show their main properties. To do so, it will

be notationally convenient to restrict ourselves to power series in z, hence

statements on asymptotic behavior of functions are always made for the

variable z tending to the origin in some region. The main applications will

be to formal analytic or meromorphic transformations, i.e., series in 1/z,

but we trust that the reader will be able to make the necessary reformula-

tion of the results.

The fact that formal analytic transformations have matrix coe¬cients

does not cause any additional problems: If we develop a theory for scalar

power series, it immediately carries over to series with matrix coe¬cients,

since these are the same as a matrix whose entries are scalar power series.

However, e¬orts have been made very recently to generalize the theory of

multisummability to partial di¬erential equations, or ordinary di¬erential

equations depending upon a parameter, and in this context one has to allow

power series with coe¬cients in some space of functions in one or more

variables. Therefore, we shall consider a ¬xed but arbitrary Banach space

E over C , which we shall frequently assume to be a Banach algebra, and we

will study functions, resp. power series, with values, resp. coe¬cients, in

E . Many of the results to follow are true even for E being a more general

space: Sometimes no topology on E is needed, so E might be an arbitrary

vector space, or an algebra, over C . Even when a topology is required, one

may verify that instead of a norm one can make do with one or several

seminorm(s) on E , hence E may be a Fr´chet space. We will, however,

e

always assume that E , equipped with some ¬xed norm · , is a Banach

60 4. Asymptotic Power Series

space. The classical case then occurs for E = C , which is a Banach algebra

under · = |·|, the modulus of complex numbers, and readers who are only

interested in this case may always specialize the de¬nitions and results by

setting E = C , or = C ν , or = C ν—ν . For a brief description of the theory

of holomorphic functions with values in a Banach space, and in particular

of those results applied in this book, refer to Appendix B.

Aside from some of the books listed in Chapter 1, the theory of asymp-

totic power series has been presented by the following other authors: Erd´lyi

e

[101], Ford [103], de Bruijn [75], Olver [212], Pittnauer [219], Dingle [89],

and Sternin and Shatalov [257].

Recently, several authors have developed a new theory under the name

of hyper-asymptotics, which is not discussed here. As references, we men-

tion [196, 209, 210].

4.1 Sectors and Sectorial Regions

We will deal with holomorphic functions, which in general have a branch

point at the origin. Therefore, it is convenient to think of these functions as

de¬ned in sectorial regions on the Riemann surface of the natural logarithm,

which is brie¬‚y described on p. 226 (Appendix B).

A sector on the Riemann surface of the logarithm will be a set of the

form S = S(d, ±, ρ) = {z : 0 < |z| < ρ, |d ’ arg z| < ±/2}, where d is an

arbitrary real number, ± is a positive real, and ρ either is a positive real

number or ∞. We shall refer to d, resp. ±, resp. ρ, as the bisecting direc-

tion, resp. the opening, resp. the radius of S. Observe that on the Riemann

surface of the logarithm the value arg z, for every z = 0, is uniquely de-

¬ned; hence the bisecting direction of a sector is uniquely determined. In

particular, if ρ = ∞, resp. ρ < ∞, we will speak of S having in¬nite, resp.

¬nite, radius. It should be kept in mind that we do not consider sectors of

in¬nite opening, nor an empty sector. If we write S(d, ±, ρ), then it shall

go without saying that d, ±, ρ are as above. In case ρ = ∞, we mostly write

S(d, ±) instead of S(d, ±, ∞). A closed sector is a set of the form

¯ ¯

S = S(d, ±, ρ) = {z : 0 < |z| ¤ ρ, |d ’ arg z| ¤ ±/2}

with d and ± as before, but ρ a positive real number, i.e., never equal to

∞. Hence closed sectors always are of ¬nite radius, and they never contain

the origin. Therefore, closed sectors are not closed as subsets of C , but are

so as subsets of the Riemann surface of the logarithm, since this surface

does not include the origin.

A region G, on the Riemann surface of the logarithm, will be named a

sectorial region, if real numbers d and ± > 0 exist such that G ‚ S(d, ±),

¯

while for every β with 0 < β < ± one can ¬nd ρ > 0 for which S(d, β, ρ) ‚

G. We shall call ± the opening and d the bisecting direction of G, and we

4.2 Functions in Sectorial Regions 61

frequently write G(d, ±) for a sectorial region of bisecting direction d and

opening ±. Observe, however, that the notation G(d, ±) does not imply that

the region is unbounded, as it would be for sectors. Also, a sectorial region

is not uniquely described by its opening and bisecting direction, but these

two will be its essential characteristics. As an example, we mention that

the open disc centered at some point z0 = 0 and having radius equal to |z0 |

is a sectorial region of opening π and bisecting direction arg z0 . For more

examples, see the following exercises.

Exercises:

1. For k > 0 and a ∈ C \ {0}, show that the mapping z ’ a z k maps

sectorial regions of opening ± and bisecting directions d to sectorial

regions of opening k± and bisecting direction kd + arg a.

2. For k > 0, c ≥ 0 and „ ∈ R, show that the set of points described by

the inequalities

k|„ ’ arg z| < π/2, cos(k[„ ’ arg z]) > c|z|k , (4.1)

is a sectorial region of opening π/k and bisecting direction „ . In par-

ticular, picture the case of c = 0. Observe that the ¬rst inequality is

needed to specify a sheet of the Riemann surface of the logarithm on

which the sectorial region is situated.

3. For ¬xed ± > 0, and d in some open interval I, let sectorial regions

G(d, ±) be given. Show that their union is again a sectorial region,

and ¬nd its opening and bisecting direction.

4.2 Functions in Sectorial Regions

Let G be a given sectorial region, and let f ∈ H(G, E ) “ hence f (z) may be

multi-valued if G has opening larger than 2π; see p. 227 for the de¬nition

of single- versus multi-valuedness. We say that f is bounded at the origin,

¯

if for every closed subsector S of G there exists a positive real constant

¯ ¯

c, depending on S, such that f (z) ¤ c for z ∈ S. We say that f is

continuous at the origin, if an element of E , denoted by f (0), exists such

that

f (z) ’’ f (0), G z ’ 0,

with convergence being uniform in every closed subsector, meaning that for

¯

every S ‚ G and every µ > 0 there exists ρ > 0 so that f (z) ’ f (0) ¤ µ

¯

for z ∈ S with |z| < ρ. Hence, continuity at the origin assures existence of

62 4. Asymptotic Power Series

a limit when approaching the origin along curves1 staying within a closed

subsector of G, while no limit may exist for general curves in G. Similarly,

statements upon continuity, resp. a limit, as z ’ ∞ are always meant to

imply that convergence is uniform in closed subsectors.

We say that f , holomorphic in some sectorial region G, is holomorphic

at the origin, if f can be holomorphically continued into a sector S ⊃ G

of opening more than 2π, and if f then is single-valued in S and bounded

at the origin; the well-known result on removable singularities then implies

that f has a convergent power series expansion about the origin; for the

notion of essential singularities, poles, and removable singularities, compare

Exercise 2 on p. 223.

We say that f is di¬erentiable at the origin, if a complex number f (0)

exists so that the quotient (f (z)’f (0))/z is continuous at the origin, in the

sense de¬ned above, and then its limit as z ’ 0 in G is denoted by f (0)

and named the derivative of f at the origin. In the exercises below, we shall

show that di¬erentiability of f at the origin is equivalent to continuity of

f there, and that f (0) is equal to the limit of f (z) as z ’ 0 in G.

Obviously, di¬erentiability implies continuity of the function at the origin,

but a function can be di¬erentiable at the origin without being holomorphic

there “ for this, compare one of the exercises below. Moreover, we say that

f is n-times di¬erentiable at the origin, if its (n ’ 1)st derivative f (n’1) (z)

is di¬erentiable at the origin. As follows from the exercises below, this is

equivalent to f (n) (z) being continuous at the origin.

Let S = S(d, ±) be a sector of in¬nite radius, and let f (z) be holomorphic

in S. Suppose that k > 0 exists for which the following holds true:

To every • with |d ’ •| < ±/2 there exist ρ, c1 , c2 > 0 such that for every

z with |z| ≥ ρ, |d ’ arg z| ¤ •,

f (z) ¤ c1 exp[c2 |z|k ].

Then we shall say that f (z) is of exponential growth at most k (in S).

This notion compares to that of (exponential) order as follows: If f (z) is of

exponential growth at most k (in S), then it either is of order less than k,

or of order equal to k and of ¬nite type, and vice versa (see the Appendix

for the de¬nition of order and type, and formulas relating both to the

coe¬cients of an entire function). The set of all functions f , holomorphic

and of exponential growth at most k in S and continuous at the origin,

shall be denoted by A(k) (S, E ). As an example, we mention Mittag-Le¬„er™s

function E± (z), de¬ned on p. 233. It is an entire function of exponential

order k = 1/± and ¬nite type (equal to one); hence it is of exponential

growth (at most) k in every sector of in¬nite radius. More generally, if

fn ∈ E , n ≥ 0, are such that for some c > 0 we have fn ¤ cn , n ≥ 0,

1 Note

that a curve is given by an arbitrary continuous mapping x(t), a ¤ t ¤ b, while

the term path means a recti¬able curve, i.e., a curve of ¬nite length.

4.2 Functions in Sectorial Regions 63

then

∞

fn z n /“(1 + n/k)

f (z) =

n=0

is bounded by E1/k (c|z|), and therefore f (z) is of exponential growth at

most k in every sector of in¬nite radius. We shall occasionally say that a

function f is of exponential growth not more than k in a direction d, if for

some µ > 0 we have f ∈ A(k) (S(d, µ), E ).

Exercises: For some of the following exercises, let E = C and de¬ne

∞ ∞

zt ’t

exp[t(z ’ log t)] dt,

g(z) = et dt = (4.2)

0 0

integrating along the positive real axis. This interesting example is from

Newman [203].

1. Let G be a sectorial region, and let f (z) be analytic in G and so that

f (z) is continuous at the origin. Let f (0) denote the limit of f (z)

as z ’ 0 in G. Show that f (z) then is di¬erentiable at the origin,

and f (0) is equal to its derivative at the origin.

2. Let G be a sectorial region, and let f (z) be analytic in G and dif-

ferentiable at the origin. Let f (0) denote the derivative of f at the

origin. Show that f (z) then is continuous at the origin, and f (0) is

equal to the limit of f (z) as z ’ 0 in G.

3. Let G be an arbitrary sectorial region, and consider f (z) = z µ log z,

for ¬xed µ ∈ C . For the largest integer k (if any) with 0 ¤ k < Re µ,

show that f is k-times, but not (k + 1)-times, di¬erentiable at the

origin.

4. Show that g(z) as in (4.2) is an entire function, and give the coe¬-

cients of its power series expansion in terms of some integral.

5. For Im z = π/2 + c, c > 0, show that Cauchy™s theorem allows in

(4.2) to replace integration along the real axis by integration along the

positive imaginary axis. Use this to show for these z that |g(z)| ¤ 1/c.

Prove the same estimate for Im z = ’(π/2 + c).

6. For every sector S of in¬nite radius, not containing the positive real

axis, conclude that g(z) is of exponential order zero in S.

7. Use Phragmen-Lindel¨f™s principle (p. 235) or a direct lower estimate

o

of g(x) for x > 0 to show that g(z) cannot be of ¬nite exponential

growth in sectors S including the positive real axis; hence g is an

entire function of in¬nite order.

64 4. Asymptotic Power Series