Differential Equations

F.G. Tricomi, Elizabeth A. McHarg

• Publisher: Dover Publications
• Year: 2013
• eBook ISBN: 9780486299297
• Topic: Mathematics
• Subtopic: Differential Equations
• Index: Mathematics

Based on his extensive experience as an educator, F. G. Tricomi wrote this practical and concise teaching text to offer a clear idea of the problems and methods of the theory of differential equations. The treatment is geared toward advanced undergraduates and graduate students and addresses only questions that can be resolved with rigor and simplicity.
Starting with a consideration of the existence and uniqueness theorem, the text advances to the behavior of the characteristics of a first-order equation, boundary problems for second-order linear equations, asymptotic methods, and differential equations in the complex field. The author discusses only ordinary differential equations, excluding coverage of the methods of integration and stressing the importance of reading the properties of the integrals directly from the equations. An extensive bibliography and helpful indexes conclude the text.

V. Differential equations in the complex field

41. Majorizing functions

So far we have been concerned exclusively with differential equations in the real field which in most cases is the natural field to use. However there are several questions—for example, that of asymptotic behaviour which has already been discussed in the preceding chapter—which are most naturally developed in the complex field, or as is sometimes said the analytic field, as the functions under consideration are supposed to be analytic, i.e. they may be represented by power series.

As a first step in introducing the complex field we require to supplement the fundamental theorem of existence of Chapter I to establish that when the differential equations of a given system are analytic then the solutions are also analytic; we therefore prove the following theorem:

Given a normal system of differential equations of the form

(1)

where f₁, f₂, . . , f_n are n analytic functions of the n + 1 complex variables x, y₁, y₂, . . . , y_n regular within the vicinity of the point

(2)

i.e. such that each function considered as a function of x may be developed in a power series (of positive integral powers) of x − x₀, considered as a function of y₁ may be developed in a power series of

, etc., where all the radii of convergence are non-zero, then there exists one and only one system of analytic functions of x regular within a certain vicinity (whose bounds can be explicitly stated), which system satisfied (1) identically and which assumes respectively the values

,

, . . . ,

for x = x₀.

This theorem may be proved by the method of successive approximations as used in Chapter I, with few modifications. It is however of real value to use a different method—a method which may also be used for systems of equations in the real field although it belongs most naturally to the complex field—originally used by Cauchy and developed earlier than the method of successive approximations; it was in fact the first method to produce rigorous resul...

Table of contents

• Title Page
• Copyright Page
• Preface to the English edition
• Preface to the first Italian edition
• Preface to the second Italian edition
• Table of Contents
• I. The existence and uniqueness theorem
• II. The behaviour of the characteristics of a first-order equation
• III. Boundary problems for linear equations of the second order
• IV. Asymptotic Methods
• V. Differential equations in the complex field
• Bibliography
• Author Index
• General Index