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Formal Power Series and
Linear Systems of
Meromorphic Ordinary
Differential Equations

Werner Balser

F¨r meine verstorbenen Eltern,

f¨r meine liebe Frau und unsere drei S¨hne.
u o

This book aims at two, essentially di¬erent, types of readers: On one hand,
there are those who have worked in, or are to some degree familiar with,
the section of mathematics that is described here. They may want to have
a source of reference to the recent results presented here, replacing my
text [21], which is no longer available, but will need little motivation to
start using this book. So they may as well skip reading this introduction,
or immediately proceed to its second part (p. v) which in some detail
describes the content of this book. On the other hand, I expect to attract
some readers, perhaps students of colleagues of the ¬rst type, who are not
familiar with the topic of the book. For those I have written the ¬rst part of
the introduction, hoping to attract their attention and make them willing
to read on.

Some Introductory Examples
What is this book about? If you want an answer in one sentence: It is con-
cerned with formal power series “ meaning power series whose radius of
convergence is equal to zero, so that at ¬rst glance they may appear as
rather meaningless objects. I hope that, after reading this book, you may
agree with me that these formal power series are fun to work with and
really important for describing some, perhaps more theoretical, features
of functions solving ordinary or partial di¬erential equations, or di¬erence
equations, or perhaps even more general functional equations, which are,
however, not discussed in this book.
viii Preface

Do such formal power series occur naturally in applications? Yes, they
do, and here are three simple examples:

ˆ n+1
1. The formal power series f (z) = 0 n! z formally satis¬es the
ordinary di¬erential equation (ODE for short)
z 2 x = x ’ z. (0.1)
But everybody knows how to solve such a simple ODE, so why care
about this divergent power series? Yes, that is true! But, given a
slightly more complicated ODE, we can no longer explicitly compute
its solutions in closed form. However, we may still be able to compute
solutions in the form of power series. In the simplest case, the ODE
may even have a solution that is a polynomial, and such solutions can
sometimes be found as follows: Take a polynomial p(z) = 0 pn z n
with undetermined degree m and coe¬cients pn , insert into the ODE,
compare coe¬cients, and use the resulting equations, which are linear
for linear ODE, to compute m and pn . In many cases, in particular
for large m, we may not be able to ¬nd the values pn explicitly. How-
ever, we may still succeed in showing that the system of equations
for the coe¬cients has one or several solutions, so that at least the
existence of polynomial solutions follows. In other cases, when the
ODE does not have polynomial solutions, one can still try to ¬nd,
or show the existence of, solutions that are “polynomials of in¬nite
degree,” meaning power series

ˆ fn (z ’ z0 )n ,
f (z) =

with suitably chosen z0 , and fn to be determined from the ODE.
While the approach at ¬rst is very much analogous to that for poly-
nomial solutions, two new problems arise: For one thing, we will get
a system of in¬nitely many equations in in¬nitely many unknowns,
namely, the coe¬cients fn ; and secondly, we are left with the prob-
lem of determining the radius of convergence of the power series. The
¬rst problem, in many cases, turns out to be relatively harmless, be-
cause the system of equations usually can be made to have the form
of a recursion: Given the coe¬cients f0 , . . . , fn , we can then compute
the next coe¬cient fn+1 . In our example (0.1), trying to compute
a power series solution f (z), with z0 = 0, immediately leads to the
identities f0 = 0, f1 = 1, and fn+1 = n fn , n ≥ 1. Even to ¬nd the
radius of convergence of the power series may be done, but as the
above example shows, it may turn out to be equal to zero!
2. Consider the di¬erence equation
x(z + 1) = (1 ’ a z ’2 ) x(z).
Preface ix

After some elementary calculations, one can show that this di¬erence

equation has a unique solution of the form f (z) = 1 + 1 fn z ’n ,
which is a power series in 1/z. The coe¬cients can be uniquely com-
puted from the recursion obtained from the di¬erence equation, and
they grow, roughly speaking, like n! so that, as in the previous case,
the radius of convergence of the power series is equal to zero. Again,
this example is so simple that one can explicitly compute its solutions
in terms of Gamma functions. But only slightly more complicated
di¬erence equations cannot be solved in closed form, while they still
have solutions in terms of formal power series.
3. Consider the following problem for the heat equation:

ut = uxx , u(0, x) = •(x),

with a function • that we assume holomorphic in some region G.

This problem has a unique solution u(t, x) = 0 un (x) tn , with co-
e¬cients given by

•(2n) (x)
n ≥ 0.
un (x) = ,
This is a power series in the variable t, whose coe¬cients are functions
of x that are holomorphic in G. As can be seen from Cauchy™s Integral
Formula, the coe¬cients un (x), for ¬xed x ∈ G, in general grow like
n! so that the power series has radius of convergence equal to zero.

So formal power series do occur naturally, but what are they good for? Well,
this is exactly what this book is about. In fact, it presents two di¬erent
but intimately related aspects of formal power series:
For one thing, the very general theory of asymptotic power series expan-
sions studies certain functions f that are holomorphic in a sector S but
singular at its vertex, and have a certain asymptotic behavior as the variable
approaches this vertex. One way of describing this behavior is by saying
that the nth derivative of the function approaches a limit fn as the variable
z, inside of S, tends toward the vertex z0 of the sector. As we shall see, this
is equivalent to saying that the function, in some sense, is in¬nitely often
di¬erentiable at z0 , without being holomorphic there, because the limit of
the quotient of di¬erences will only exist once we stay inside of the sector.
The values fn may be regarded as the coe¬cients of Taylor™s series of f ,
but this series may not converge, and even when it does, it may not con-
verge toward the function f . Perhaps the simplest example of this kind is
the function f (z) = e’1/z , whose derivatives all tend to fn = 0 whenever z
tends toward the origin in the right half-plane. This also shows that, unlike
for functions that are holomorphic at z0 , this Taylor series alone does not
determine the function f . In fact, given any sector S, every formal power
series f arises as an asymptotic expansion of some f that is holomorphic
x Preface

in S, but this f never is uniquely determined by f , so that in particular the
value of the function at a given point z = z0 in general cannot be computed
from the asymptotic power series. In this book, the theory of asymptotic
power series expansions is presented, not only for the case when the co-
e¬cients are numbers, but also for series whose coe¬cients are in a given
Banach space. This generalization is strongly motivated by the third of the
above examples.
While general formal power series do not determine one function, some
of them, especially the ones arising as solutions of ODE, are almost as well-
behaved as convergent ones: One can, more or less explicitly, compute some
function f from the divergent power series f , which in a certain sector
is asymptotic to f . In addition, this function f has other very natural
properties; e.g., it satis¬es the same ODE as f . This theory of summability
of formal power series has been developed very recently and is the main
reason why this book was written.
If you want to have a simple example of how to compute a function from

a divergent power series, take f (z) = 0 fn z n , assuming that |fn | ¤ n!
for n ≥ 0. Dividing the coe¬cients by n! we obtain a new series converging
at least for |z| < 1. Let g(z) denote its sum, so g is holomorphic in the unit
disc. Now the general idea is to de¬ne the integral

g(u) e’u/z du
f (z) = z (0.2)

as the sum of the series f . One reason for this to be a suitable de¬nition is
the fact that if we replace the function g by its power series and integrate
termwise (which is illegal in general), then we end up with f (z). While
this motivation may appear relatively weak, it will become clear later that
this nonetheless is an excellent de¬nition for a function f deserving the
title sum of f “ except that the integral (0.2) may not make sense for one
of the following two reasons: The function g is holomorphic in the unit
disc but may be unde¬ned for values u with 1 u ≥ 1, making the integral
entirely meaningless. But even if we assume that g can be holomorphi-
cally continued along the positive real axis, its rate of growth at in¬nity
may be such that the integral diverges. So you see that there are some
reasons that keep us from getting a meaningful sum for f in this simple
fashion, and therefore we shall have to consider more complicated ways of
summing formal power series. Here we shall present a summation process,
called multisummability, that can handle every formal power series which
solves an ODE, but is still not general enough for solutions of certain dif-
ference equations or partial di¬erential equations. Jean Ecalle, the founder
of the theory of multisummability, has also outlined some more general

1 Observe
that such an inequality should be understood as saying: Here, the number
u must be real and at least 1.
Preface xi

summation methods suitable for di¬erence equations, but we shall not be
concerned with these here.

Content of this Book
This book attempts to present the theory of linear ordinary di¬erential
equations in the complex domain from the new perspective of multisumma-
bility. It also brie¬‚y describes recent e¬orts on developing an analogous
theory for nonlinear systems, systems of di¬erence equations, partial dif-
ferential equations, and singular perturbation problems. While the case of
linear systems may be said to be very well understood by now, much more
needs to be done in the other cases.
The material of the book is organized as follows: The ¬rst two chapters
contain entirely classical results on the structure of solutions near regular,
resp.2 regular-singular, points. They are included here mainly for the sake
of completeness, since none of the problems that the theory of multisumma-
bility is concerned with arise in these cases. A reader with some background
on ODE in the complex domain may very well skip these and immediately
advance to Chapter 3, where we begin discussing the local theory of systems
near an irregular singularity. Classically, this theory starts with showing ex-
istence of formal fundamental solutions, which in our terminology will turn
out to be multisummable, but not k-summable, for any k > 0. So in a way,
these classical formal fundamental solutions are relatively complicated ob-
jects. Therefore, we will in Chapter 3 introduce a di¬erent kind of what we
shall call highest level formal fundamental solutions, which have much bet-
ter theoretical properties, although they are somewhat harder to compute.
In the following chapters we then present the theory of asymptotic power
series with special emphasis on Gevrey asymptotics and k-summability.
In contrast to the presentation in [21], we here treat power series with
coe¬cients in a Banach space. The motivation for this general approach
lies in applications to PDE and singular perturbation problems that shall
be discussed brie¬‚y later. A reader who is not interested in this general
setting may concentrate on series with coe¬cients in the complex number
¬eld, but the general case really is not much more di¬cult.
In Chapters 8 and 9 we then return to the theory of ODE and discuss the
Stokes phenomenon of highest level. Here it is best seen that the approach
we take here, relying on highest level formal fundamental solutions, gives a
far better insight into the structure of the Stokes phenomenon, because it
avoids mixing the phenomena occurring on di¬erent levels. Nonetheless, we
then present the theory of multisummability in the following chapters and
indicate that the classical formal fundamental solutions are indeed multi-
summable. The remaining chapters of the book are devoted to related but

2 Short for “respectively.”
xii Preface

di¬erent problems such as Birkho¬ ™s reduction problem or applications of
the theory of multisummability to di¬erence equations, partial di¬erential
equations, or singular perturbation problems. Several appendices provide
the results from other areas of mathematics required in the book; in par-
ticular some well-known theorems from the theory of complex variables are
presented in the more general setting of functions with values in a Banach
The book should be readable for students and scientists with some back-
ground in matrix theory and complex analysis, but I have attempted to
include all the (nonelementary) results from these areas in the appendices.

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