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2020 | Book

Introduction Read chapter Read first chapter

Linear Differential Equations in the Complex Domain

From Classical Theory to Forefront

Author: Prof. Dr. Yoshishige Haraoka

Publisher: Springer International Publishing

Book Series : Lecture Notes in Mathematics

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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About this book

This book provides a detailed introduction to recent developments in the theory of linear differential systems and integrable total differential systems. Starting from the basic theory of linear ordinary differential equations and integrable systems, it proceeds to describe Katz theory and its applications, extending it to the case of several variables. In addition, connection problems, deformation theory, and the theory of integral representations are comprehensively covered. Complete proofs are given, offering the reader a precise account of the classical and modern theory of linear differential equations in the complex domain, including an exposition of Pfaffian systems and their monodromy problems. The prerequisites are a course in complex analysis and the basics of differential equations, topology and differential geometry. This book will be useful for graduate students, specialists in differential equations, and for non-specialists who want to use differential equations.

Table of Contents

Frontmatter
Chapter 1. Introduction
Abstract
In this book, we start from a basic theory of linear ordinary differential equations in the complex domain, arrive at Katz–Oshima theory, which is the forefront, and then explain the deformation theory and the theory of completely integrable systems in several variables so that the reader will get sufficient knowledge. In the following, we illustrate the contents of this book by using several examples.
Yoshishige Haraoka

Ordinary Differential Equations

Frontmatter
Chapter 2. Scalar Differential Equations and Systems of Differential Equations
Abstract
Linear ordinary differential equations are usually given in the scalar form
Yoshishige Haraoka
Chapter 3. Analysis at a Regular Point
Abstract
We consider a system of differential equations
Yoshishige Haraoka
Chapter 4. Regular Singular Points
Abstract
Let a be a point in \(\mathbb {C}\), and f(x) a (multi-valued) holomorphic function in a neighborhood of a except a. The point x = a is said to be a regular singular point of f(x) if f(x) is not holomorphic at x = a, and if there exists a positive number N such that, for any θ 1 < θ 2,
Yoshishige Haraoka
Chapter 5. Monodromy
Abstract
In the last section we have considered the local monodromy, which describes the change of the fundamental system of solutions caused by the analytic continuation along a loop encircling a regular singular point. In this chapter we study the (global) monodromy, which describes the changes caused by global analytic continuations.
Yoshishige Haraoka
Chapter 6. Connection Problem
Abstract
We start with considering a vibration of a string. Let a string of length ℓ be set on the interval [0, ℓ] of x-axis in xy-plane. Then we can regard the vibration of the string as a motion of each point on x ∈ [0, ℓ] in y-direction. We denote the deviation in y-direction of a point at x at the time t by y(t, x). Then the function y(t, x) in two variables represents the vibration.
Yoshishige Haraoka
Chapter 7. Fuchsian Differential Equations
Abstract
As we defined in Chap. 5 (Definition 5.​2), a differential equation with coefficient in rational functions is called Fuchsian if all singular points are regular singular points. Note that x = ∞ is a regular singular point if, for a local coordinate t at \(\infty \in \mathbb {P}^1\), t = 0 is a regular singular point.
Yoshishige Haraoka
Chapter 8. Deformation Theory
Abstract
We consider a system of Fuchsian ordinary differential equations
$$\displaystyle \frac {dY}{dx}=A(x)Y, $$
where A(x) is an n × n-matrix with rational functions in x as entries.
Yoshishige Haraoka
Chapter 9. Integral Representations of Solutions of Euler Type
Abstract
Some differential equations have integral representations of solutions of Euler type. A typical example is the Gauss hypergeometric differential equation. The study of such integrals is one of the main topics in differential equations in the complex domain, however, we do not develop the general theory of integrals of Euler type. Instead, we shall study several illustrative examples, which will make the readers understand the main notions and main ideas.
Yoshishige Haraoka
Chapter 10. Irregular Singular Points
Abstract
For a linear differential equation, a pole of the coefficient of the equation is called an irregular singular point if it is not a regular singular point. In this chapter, we explain the outline of the theory of irregular singular point. Several assertions are proved.
Yoshishige Haraoka

Completely Integrable Systems

Frontmatter
Chapter 11. Linear Pfaffian Systems and Integrability Condition
Abstract
We denote by x = (x 1, x 2, …, x n) the coordinate of \(\mathbb {C}^n\). Let N be an integer. Let \(X\subset \mathbb {C}^n\) be a domain, and \(a_{ij}^k(x)\) (1 ≤ i, j ≤ N, 1 ≤ k ≤ n) be holomorphic functions on X. The system
$$\displaystyle \frac {\partial u_i}{\partial x_k}=\sum _{j=1}^Na_{ij}^k(x)u_j \quad (1\leq i\leq N,\,1\leq k\leq n) $$
of partial differential equations of the first order with unknown functions u 1(x), u 2(x), …, u N(x) is called a linear Pfaffian system.
Yoshishige Haraoka
Chapter 12. Regular Singularity
Abstract
In this section, we study the behavior of solutions of a linear Pfaffian system in a neighborhood of a singular point of the coefficients, where we restrict singular points to ones corresponding to regular singular points in ordinary differential equations. Note that a singular point of analytic functions in several variables is not a point but a set of codimension one (a hypersurface).
Yoshishige Haraoka
Chapter 13. Monodromy Representations
Abstract
We have shown in Theorem 11.​2 that, for a completely integrable linear Pfaffian system, any solution can be analytically continued to the full domain X of definition of the coefficients. Then, taking any point b ∈ X and a fundamental system of solutions \(\mathcal {U}(x)\) in its simply connected neighborhood, we have the monodromy representation.
Yoshishige Haraoka
Chapter 14. Middle Convolution
Abstract
We have seen that the middle convolution plays a substantial role in the study of Fuchsian ordinary differential equations. Then it is natural to extend the middle convolution to completely integrable systems. In this chapter, we explain how to extend the middle convolution to completely integrable systems, and how to describe it. As an illustrative example, we give an explicit description of the middle convolution for linear Pfaffian systems of KZ (Knizhnik-Zamolodchikov) type appeared in Example 11.​2 in Chap. 11. We also study the properties and applications of our middle convolution.
Yoshishige Haraoka
Backmatter
Metadata
Title
Linear Differential Equations in the Complex Domain
Author
Prof. Dr. Yoshishige Haraoka
Copyright Year
2020
Electronic ISBN
978-3-030-54663-2
Print ISBN
978-3-030-54662-5
DOI
https://doi.org/10.1007/978-3-030-54663-2

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