Elliptic Integrals and Elliptic Functions / by Takashi Takebe.

Format:

Book

Author/Creator:

Takebe, Takashi, author.

Series:

Moscow Lectures, 2522-0322 ; 9

Language:

English

Subjects (All):

Functions, Special.

Functions of complex variables.

Mathematical physics.

Special Functions.

Functions of a Complex Variable.

Mathematical Methods in Physics.

Local Subjects:

Special Functions.

Functions of a Complex Variable.

Mathematical Methods in Physics.

Physical Description:

1 online resource (329 pages)

Edition:

1st ed. 2023.

Place of Publication:

Cham : Springer International Publishing : Imprint: Springer, 2023.

Summary:

This book gives a comprehensive introduction to those parts of the theory of elliptic integrals and elliptic functions which provide illuminating examples in complex analysis, but which are not often covered in regular university courses. These examples form prototypes of major ideas in modern mathematics and were a driving force of the subject in the eighteenth and nineteenth centuries. In addition to giving an account of the main topics of the theory, the book also describes many applications, both in mathematics and in physics. For the reader’s convenience, all necessary preliminaries on basic notions such as Riemann surfaces are explained to a level sufficient to read the book. For each notion a clear motivation is given for its study, answering the question ‘Why do we consider such objects?’, and the theory is developed in a natural way that mirrors its historical development (e.g., ‘If there is such and such an object, then you would surely expect this one’). This feature sets this text apart from other books on the same theme, which are usually presented in a different order. Throughout, the concepts are augmented and clarified by numerous illustrations. Suitable for undergraduate and graduate students of mathematics, the book will also be of interest to researchers who are not familiar with elliptic functions and integrals, as well as math enthusiasts. .

Contents:

Introduction

Chapter 1. The arc length of curves

Chapter 2. Classification of elliptic integrals

Chapter 3. Applications of elliptic integrals

Chapter 4. Jacobi’s elliptic functions on R

Chapter 5. Applications of Jacobi’s elliptic functions

Riemann surfaces of algebraic functions

Chapter 7. Elliptic curves

Chapter 8. Complex elliptic integrals

Chapter 9. Mapping the upper half plane to a rectangle

Chapter 10. The Abel-Jacobi theorem

Chapter 11. The general theory of elliptic functions

Chapter 12. The Weierstrass ℘-function

Chapter 13. Addition theorems

Chapter 14. Characterisation by addition formulae

Chapter 15. Theta functions

Chapter 16. Infinite product factorisation of theta functions

Chapter 17. Complex Jacobian functions

Appendix A. Theorems in analysis and complex analysis

Bibliography

Index.

Notes:

Includes bibliographical references and index.

ISBN:

3-031-30265-6