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to compute corresponding convergent systems; of these systems the
¬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

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
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.

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
eigenvalues. Verify that the matrix T (z) in the formal fundamental
solution obtained in Exercise 4 on p. 45 is an HLFFS.
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,
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
[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.

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
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


fn z n /“(1 + n/k)
f (z) =

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

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
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


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