Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations

Preface
	Some Introductory Examples
	Content of this Book
	Personal Remarks
Contents
1. Basic Properties of Solutions
	1.1 Simply Connected Regions
	1.2 Fundamental Solutions
	1.3 Systems in General Regions
	1.4 Inhomogeneous Systems
	1.5 Reduced Systems
	1.6 Some Additional Notation
2. Singularities of First Kind
	2.1 Systems with Good Spectrum
	2.2 Confluent Hypergeometric Systems
	2.3 Hypergeometric Systems
	2.4 Systems with General Spectrum
	2.5 Scalar Higher-Order Equations
3. Highest-Level Formal Solutions
	3.1 Formal Transformations
	3.2 The Splitting Lemma
	3.3 Nilpotent Leading Term
	3.4 Transformation to Rational Form
	3.5 Highest-Level Formal Solutions
4. Asymptotic Power Series
	4.1 Sectors and Sectorial Regions
	4.2 Functions in Sectorial Regions
	4.3 Formal Power Series
	4.4 Asymptotic Expansions
	4.5 Gevrey Asymptotics
	4.6 Gevrey Asymptotics in Narrow Regions
	4.7 Gevrey Asymptotics in Wide Regions
5. Integral Operators
	5.1 Laplace Operators
	5.2 Borel Operators
	5.3 Inversion Formulas
	5.4 A Different Representation for Borel Operators
	5.5 General Integral Operators
	5.6 Kernels of Small Order
	5.7 Properties of the Integral Operators
	5.8 Convolution of Kernels
6. Summable Power Series
	6.1 Gevrey Asymptotics and Laplace Transform
	6.2 Summability in a Direction
	6.3 Algebra Properties
	6.4 Definition of k-Summability
	6.5 General Moment Summability
	6.6 Factorial Series
7. Cauchy-Heine Transform
	7.1 Definition and Basic Properties
	7.2 Normal Coverings
	7.3 Decomposition Theorems
	7.4 Functions with a Gevrey Asymptotic
8. Solutions of Highest Level
	8.1 The Improved Splitting Lemma
	8.2 More on Transformation to Rational Form
	8.3 Summability of Highest-Level Formal Solutions
	8.4 Factorization of Formal Fundamental Solutions
	8.5 Definition of Highest-Level Normal Solutions
9. Stokes’ Phenomenon
	9.1 Highest-Level Stokes’ Multipliers
	9.2 The Periodicity Relation
	9.3 The Associated Functions
	9.4 An Inversion Formula
	9.5 Computation of the Stokes Multipliers
	9.6 Highest-Level Invariants
	9.7 The Freedom of the Highest-Level Invariants
10. Multisummable Power Series
	10.1 Convolution Versus Iteration of Operators
	10.2 Multisummability in Directions
	10.3 Elementary Properties
	10.4 The Main Decomposition Result
	10.5 Some Rules for Multisummable Power Series
	10.6 Singular Multidirections
	10.7 Applications of Cauchy-Heine Transforms
	10.8 Optimal Summability Types
11. Ecalle’s Acceleration Operators
	11.1 Definition of the Acceleration Operators
	11.2 Ecalle’s Definition of Multisummability
	11.3 Convolutions
	11.4 Convolution Equations
12. Other Related Questions
	12.1 Matrix Methods and Multisummability
	12.2 The Method of Reduction of Rank
	12.3 The Riemann-Hilbert Problem
	12.4 Birkhoff’s Reduction Problem
	12.5 Central Connection Problems
13. Applications in Other Areas, and Computer Algebra
	13.1 Nonlinear Systems of ODE
	13.2 Difference Equations
	13.3 Singular Perturbations
	13.4 Partial Differential Equations
	13.5 Computer Algebra Methods
14. Some Historical Remarks
Appendix A. Matrices and Vector Spaces
	A.1 Matrix Equations
	A.2 Blocked Matrices
	A.3 Some Functional Analysis
Appendix B. Functions with Values in Banach Spaces
	B.1 Cauchy’s Theorem and its Consequences
	B.2 Power Series
	B.3 Holomorphic Continuation
	B.4 Order and Type of Holomorphic Functions
	B.5 The Phragmén-Lindelöf Principle
Appendix C. Functions of a Matrix
	C.1 Exponential of a Matrix
	C.2 Logarithms of a Matrix
Solutions to the Exercises
Section 1.1
Section 1.2
Section 1.3
Section 1.5
Section 2.1
Section 2.3
Section 2.5
Section 3.3
Section 4.1
Section 4.3
Section 4.7
Section 5.4
Section 5.8
Section 6.4
Section 6.6
Section 8.2
Section 8.5
Section 9.3
Section 10.1
Section 10.7
Section 12.4
Section B.1
Section B.4
Section C.2
References
1-9
10-23
24-36
37-48
49-62
63-76
77-89
90-104
105-118
119-133
134-146
147-158
159-171
172-185
186-198
199-211
212-225
226-239
240-251
252-265
266-279
280-290
Index
List of Symbols