A Brief Introduction to Theta Functions

Richard Bellman

• Publisher: Dover Publications
• Year: 2013
• Print ISBN: 9780486492957
• eBook ISBN: 9780486782836
• Topic: Mathematics
• Subtopic: Applied Mathematics
• Index: Mathematics

Brief but intriguing, this monograph on the theory of elliptic functions was written by one of America's most prominent and widely read mathematicians. Richard Bellman encompasses a wealth of material in a succession of short chapters, spotlighting high points of the fundamental regions of elliptic functions and illustrating powerful and versatile analytic methods.
Suitable for advanced undergraduates and graduate students in mathematics, this introductory treatment is largely self-contained. Topics include Fourier series, sufficient conditions, the Laplace transform, results of Doetsch and Kober-Erdelyi, Gaussian sums, and Euler's formulas and functional equations. Additional subjects include partial fractions, mock theta functions, Hermite's method, convergence proof, elementary functional relations, multidimensional Poisson summation formula, the modular transformation, and many other areas.

1. Introduction

Let us begin by introducing the functions whose many properties we shall study. Consider the function of z defined by the infinite series

Here z is a complex variable, permitted to assume any value, while t is a complex parameter satisfying the condition Re(t) > 0. It is easy to see that the series converges absolutely and uniformly in any bounded region of the z plane, and thus that f(z) is an entire function of z. It is clear, furthermore, that f(z) is periodic with period π,

Despite its rather special form, f(z) is no artificially concocted function. On the contrary, it is one of the basic functions of analysis, a theta function. It occurs in a crucial role in the development of the theory of linear partial differential equations of parabolic type, in the study of the Riemann zeta function and in connection with the representation of numbers as sums of squares, thus occupying a major position in the analytic theory of numbers, and, in its most fundamental role, it is a keystone of the theory of elliptic functions.

Jacobi, in his magnum opus, Fundamenta Nova. . ., was the first mathematician to study these functions in a thorough fashion. True to the spirit of his time, a spirit compounded of equal parts of faith and nearly incredible ingenuity, he derived his many elegant and intricate results by means of the algebraic manipulations familiar to Euler and Gauss. Once exhibited, the results are readily verified by means of the theory of functions of a complex variable, using Liouville’s theorem. Moreover, as we shall see, we now possess a number of powerful techniques for deriving results of this nature. Some are valid only for functions of one complex variable, while others can be used for functions of many variables.

Despite our unqualified title, our aim is not so ambitious as to present any complete, or even partially complete theory of theta functions. This domain is of such magnitude and extent that it cannot be mapped in any brief or simple fashion. Instead, we wish to use three principal results in the theory of elliptic functions, all expressible in terms of theta functions, as a stage upon which to parade some of the general factota of analysis, and as an excuse to discuss some intimately related results of great mathematical elegance. Our aim is to indicate the applicability and versatility of analytic techniques that should be part of the hope chest of every young mathematician.

Consequently, we shall neither follow the ingenious derivations...

Table of contents

• Cover
• Title Page
• Copyright Page
• Didication
• Foreword
• Contents
• 1. Introduction
• Index