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A reader who is mainly interested in the asymptotic theory and/or multi-
summability may leave out the beginning chapters and start reading with
Chapters 4 through 7, and then go on to Chapter 10 “ these are pretty
much independent of the others in between and may be a good basis for a
course on the subject of asymptotic power series, although the remaining
ones may provide an excellent motivation for such a general theory to be
developed.


Personal Remarks
Some personal remarks may be justi¬ed here: In fall of 1970, I came to the
newly founded University of Ulm to work under the direction of Alexan-
der Peyerimho¬ in summability theory. About 1975 I switched ¬elds and,
jointly with W. B. Jurkat and D. A. Lutz, began my studies in the very
classical, yet still highly active, ¬eld of systems of ordinary linear di¬eren-
tial equations whose coe¬cients are meromorphic functions of a complex
variable (for short: meromorphic systems of ODE). This ¬eld has occupied
most of my (mathematical) energies, until almost twenty years later when
I took up summability again to apply its techniques to the divergent power
series that arise as formal solutions of meromorphic ODE. In this book, I
have made an e¬ort to represent the classical theory of meromorphic sys-
tems of ODE in the new light shed upon it by the recent achievements in
the theory of summability of formal power series.
After more than twenty years of research, I have become highly addicted
to this ¬eld. I like it so much because it gives us a splendid opportunity
to obtain signi¬cant results using standard techniques from the theory of
complex variables, together with some matrix algebra and other classical
areas of analysis, such as summability theory, and I hope that this book
may infect others with the same enthusiasm for this fascinating area of
mathematics. While one may also achieve useful results using more sophis-
ticated tools borrowed from advanced algebra, or functional analysis, such
will not be required to understand the content of this book.
I should like to make the following acknowledgments: I am indebted to
the group of colleagues at Grenoble University, especially J. DellaDora, F.
Jung, and M. Barkatou and his students. During my appointment as Pro-
Preface xiii

fesseur Invit´ in September 1997 and February 1998, they introduced me
e
to the realm of computer algebra and helped me prepare the corresponding
section, and in addition created a perfect environment for writing a large
portion of the book while I was there. In March 1998, while I was sim-
ilarly visiting Lille University, Anne Duval made me appreciate the very
recent progress on application of multisummability to the theory of di¬er-
ence equations, for which I am grateful as well. I would also like to thank
many other colleagues for support in collecting the numerous references
I added, and for introducing me to related, yet di¬erent, applications of
multisummability on formal solutions of partial di¬erential equations and
singular perturbation problems. Last, but not least, I owe thanks to my
two teachers at Ulm University, Peyerimho¬ (who died all too suddenly
in 1996) and Jurkat, who were not actively involved in writing, but from
whom I acquired the mathematics, as well as the necessary stamina, to
complete this book.
Contents




1 Basic Properties of Solutions 1
1.1 Simply Connected Regions . . . . . . . . . . . . . . . . . . . 2
1.2 Fundamental Solutions . . . . . . . . . . . . . . . . . . . . . 5
1.3 Systems in General Regions . . . . . . . . . . . . . . . . . . 8
1.4 Inhomogeneous Systems . . . . . . . . . . . . . . . . . . . . 10
1.5 Reduced Systems . . . . . . . . . . . . . . . . . . . . . . . . 12
1.6 Some Additional Notation . . . . . . . . . . . . . . . . . . . 14

2 Singularities of First Kind 17
2.1 Systems with Good Spectrum . . . . . . . . . . . . . . . . . 19
2.2 Con¬‚uent Hypergeometric Systems . . . . . . . . . . . . . . 21
2.3 Hypergeometric Systems . . . . . . . . . . . . . . . . . . . . 25
2.4 Systems with General Spectrum . . . . . . . . . . . . . . . . 27
2.5 Scalar Higher-Order Equations . . . . . . . . . . . . . . . . 34

3 Highest-Level Formal Solutions 37
3.1 Formal Transformations . . . . . . . . . . . . . . . . . . . . 38
3.2 The Splitting Lemma . . . . . . . . . . . . . . . . . . . . . . 42
3.3 Nilpotent Leading Term . . . . . . . . . . . . . . . . . . . . 45
3.4 Transformation to Rational Form . . . . . . . . . . . . . . . 52
3.5 Highest-Level Formal Solutions . . . . . . . . . . . . . . . . 55

4 Asymptotic Power Series 59
4.1 Sectors and Sectorial Regions . . . . . . . . . . . . . . . . . 60
xvi Contents

4.2 Functions in Sectorial Regions . . . . . . . . . . . . . . . . . 61
4.3 Formal Power Series . . . . . . . . . . . . . . . . . . . . . . 64
4.4 Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . 65
4.5 Gevrey Asymptotics . . . . . . . . . . . . . . . . . . . . . . 70
4.6 Gevrey Asymptotics in Narrow Regions . . . . . . . . . . . 73
4.7 Gevrey Asymptotics in Wide Regions . . . . . . . . . . . . 75

5 Integral Operators 77
5.1 Laplace Operators . . . . . . . . . . . . . . . . . . . . . . . 78
5.2 Borel Operators . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.3 Inversion Formulas . . . . . . . . . . . . . . . . . . . . . . . 82
5.4 A Di¬erent Representation for Borel Operators . . . . . . . 83
5.5 General Integral Operators . . . . . . . . . . . . . . . . . . 85
5.6 Kernels of Small Order . . . . . . . . . . . . . . . . . . . . . 89
5.7 Properties of the Integral Operators . . . . . . . . . . . . . 91
5.8 Convolution of Kernels . . . . . . . . . . . . . . . . . . . . . 93

6 Summable Power Series 97
6.1 Gevrey Asymptotics and Laplace Transform . . . . . . . . . 99
6.2 Summability in a Direction . . . . . . . . . . . . . . . . . . 100
6.3 Algebra Properties . . . . . . . . . . . . . . . . . . . . . . . 102
6.4 De¬nition of k-Summability . . . . . . . . . . . . . . . . . . 104
6.5 General Moment Summability . . . . . . . . . . . . . . . . . 107
6.6 Factorial Series . . . . . . . . . . . . . . . . . . . . . . . . . 110

7 Cauchy-Heine Transform 115
7.1 De¬nition and Basic Properties . . . . . . . . . . . . . . . . 116
7.2 Normal Coverings . . . . . . . . . . . . . . . . . . . . . . . . 118
7.3 Decomposition Theorems . . . . . . . . . . . . . . . . . . . 119
7.4 Functions with a Gevrey Asymptotic . . . . . . . . . . . . . 121

8 Solutions of Highest Level 123
8.1 The Improved Splitting Lemma . . . . . . . . . . . . . . . . 124
8.2 More on Transformation to Rational Form . . . . . . . . . . 127
8.3 Summability of Highest-Level Formal Solutions . . . . . . . 129
8.4 Factorization of Formal Fundamental Solutions . . . . . . . 131
8.5 De¬nition of Highest-Level Normal Solutions . . . . . . . . 137

9 Stokes™ Phenomenon 139
9.1 Highest-Level Stokes™ Multipliers . . . . . . . . . . . . . . . 140
9.2 The Periodicity Relation . . . . . . . . . . . . . . . . . . . . 142
9.3 The Associated Functions . . . . . . . . . . . . . . . . . . . 144
9.4 An Inversion Formula . . . . . . . . . . . . . . . . . . . . . 150
9.5 Computation of the Stokes Multipliers . . . . . . . . . . . . 151
9.6 Highest-Level Invariants . . . . . . . . . . . . . . . . . . . . 153
Contents xvii

9.7 The Freedom of the Highest-Level Invariants . . . . . . . . 155

10 Multisummable Power Series 159
10.1 Convolution Versus Iteration of Operators . . . . . . . . . . 160
10.2 Multisummability in Directions . . . . . . . . . . . . . . . . 161
10.3 Elementary Properties . . . . . . . . . . . . . . . . . . . . . 162
10.4 The Main Decomposition Result . . . . . . . . . . . . . . . 164
10.5 Some Rules for Multisummable Power Series . . . . . . . . 166
10.6 Singular Multidirections . . . . . . . . . . . . . . . . . . . . 167
10.7 Applications of Cauchy-Heine Transforms . . . . . . . . . . 169
10.8 Optimal Summability Types . . . . . . . . . . . . . . . . . . 173

11 Ecalle™s Acceleration Operators 175
11.1 De¬nition of the Acceleration Operators . . . . . . . . . . . 176
11.2 Ecalle™s De¬nition of Multisummability . . . . . . . . . . . 177
11.3 Convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . 178
11.4 Convolution Equations . . . . . . . . . . . . . . . . . . . . . 181

12 Other Related Questions 183
12.1 Matrix Methods and Multisummability . . . . . . . . . . . 184
12.2 The Method of Reduction of Rank . . . . . . . . . . . . . . 187
12.3 The Riemann-Hilbert Problem . . . . . . . . . . . . . . . . 188
12.4 Birkhoªs Reduction Problem . . . . . . . . . . . . . . . . . 189
12.5 Central Connection Problems . . . . . . . . . . . . . . . . . 193

13 Applications in Other Areas, and Computer Algebra 197
13.1 Nonlinear Systems of ODE . . . . . . . . . . . . . . . . . . 198
13.2 Di¬erence Equations . . . . . . . . . . . . . . . . . . . . . . 199
13.3 Singular Perturbations . . . . . . . . . . . . . . . . . . . . . 201
13.4 Partial Di¬erential Equations . . . . . . . . . . . . . . . . . 202
13.5 Computer Algebra Methods . . . . . . . . . . . . . . . . . . 204

14 Some Historical Remarks 207

A Matrices and Vector Spaces 211
A.1 Matrix Equations . . . . . . . . . . . . . . . . . . . . . . . . 212
A.2 Blocked Matrices . . . . . . . . . . . . . . . . . . . . . . . . 214
A.3 Some Functional Analysis . . . . . . . . . . . . . . . . . . . 215

B Functions with Values in Banach Spaces 219
B.1 Cauchy™s Theorem and its Consequences . . . . . . . . . . . 220
B.2 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
B.3 Holomorphic Continuation . . . . . . . . . . . . . . . . . . . 224
B.4 Order and Type of Holomorphic Functions . . . . . . . . . . 232
B.5 The Phragm´n-Lindel¨f Principle . . . . . .
e o . . . . . . . . . 234
xviii Contents

C Functions of a Matrix 237
C.1 Exponential of a Matrix . . . . . . . . . . . . . . . . . . . . 238
C.2 Logarithms of a Matrix . . . . . . . . . . . . . . . . . . . . 240

Solutions to the Exercises 243

References 267

Index 289

List of Symbols 295
1
Basic Properties of Solutions




In this ¬rst chapter, we discuss some basic properties of linear systems of
ordinary di¬erential equations having a coe¬cient matrix whose entries are
holomorphic functions in some region G. A reader who is familiar with the
theory of systems whose coe¬cient matrix is constant, or consists of con-
tinuous functions on a real interval, will see that all of what we say here
for the case of a simply connected region G, i.e., a region “without holes,”
is quite analogous to the real-variable situation, but we shall discover a
new phenomenon in case of multiply connected G. While for simply con-
nected regions solutions always are holomorphic in the whole region G, this
will no longer be true for multiply connected ones: Solutions will be lo-
cally holomorphic, i.e., holomorphic on every disc contained in G. Globally,
however, they will in general be multivalued functions that should best be
considered on some Riemann surface over G. As an example, observe that
x(z) = z(1 ’ z) is a solution of the equation

1 ’ 2z
z ∈ G = C \ {0, 1};
x= x,
2z (1 ’ z)

this solution has branch-points at z = 1 and the origin. Luckily, we shall
have no need to study this monodromy behavior of solutions for a general
multiply connected region. Instead, it will be su¬cient to consider the sim-
plest type of such regions, namely punctured discs. Assuming for simplicity
that we have a punctured disc about the origin, the corresponding Riemann
surface “ or to be exact, the universal covering surface “ is the Riemann
surface of the (natural) logarithm. We require the reader to have some in-
2 1. Basic Properties of Solutions

tuitive understanding of this concept, but we shall also discuss this surface
on p. 226 in the Appendix.
Most of the time we shall restrict ourselves to systems of ¬rst-order
linear equations. Since every νth order equation can be rewritten as a
system (see Exercise 5 on p. 4), our results carry over to such equations as
well. However, in some circumstances scalar equations are easier to handle
than systems. So for practical purposes, such as computing power series
solutions, we do not recommend to turn a given scalar equation into a
system, but instead one should work with the scalar equation directly.
Many books on ordinary di¬erential equations contain at least a chapter
or two dealing with ODE in the complex plane. Aside from the books of
Sibuya and Wasow, already mentioned in the introduction, we list the fol-
lowing more recent books in chronological order: Ince [138], Bieberbach [52],
Sch¨fke and Schmidt [236], and Hille [120].
a


1.1 Simply Connected Regions
Throughout this chapter, we consider a system of the form

z ∈ G,
x = A(z) x, (1.1)

where A(z) = [akj (z)] denotes a ν —ν matrix whose entries are holomorphic
functions in some ¬xed region G ‚ C , which we here assume to be simply
connected. It is notationally convenient to think of such a matrix A(z) as
a holomorphic matrix-valued function in G.
Since we know from the theory of functions of a complex variable that
such functions, if (once) di¬erentiable in an open set, are automatically
holomorphic there, it is obvious that solutions x(z) of (1.1) are always
vector-valued holomorphic functions. However, it is not clear o¬-hand that
a solution always is holomorphic in all of the region G, but we shall prove
this here. To begin, we show the following weaker result, which holds for
arbitrary regions G.
Lemma 1 Let a system (1.1), with A(z) holomorphic in a region G ‚ C ,
be given. Then for every z0 ∈ G and every x0 ∈ C ν , there exists a unique
vector-valued function x(z), holomorphic in the largest disc D = D(z0 , ρ) =
{z : |z ’ z0 | < ρ } contained in G, such that

z ∈ D,
x (z) = A(z) x(z), x(z0 ) = x0 .

Hence we may say for short that every initial value problem has a unique
solution that is holomorphic near z0 .

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