Complex Analysis and Special Functions : Cauchy Formula, Elliptic Functions and Laplace's Method.

Format:

Book

Author/Creator:

Serov, Valery.

Contributor:

Harju, Markus.

Series:

De Gruyter Textbook Series

Language:

English

Subjects (All):

Orthogonal polynomials.

Elliptic functions.

Physical Description:

1 online resource (362 pages)

Edition:

1st ed.

Place of Publication:

Berlin/Boston : Walter de Gruyter GmbH, 2025.

Summary:

This book, authored by Valery Serov and Markus Harju, serves as a comprehensive university-level textbook on complex analysis and special functions, intended for mathematics and physics students. It covers fundamental topics in complex analysis, including analytic functions, Cauchy theorem, conformal mappings, and residue theory, while also delving into advanced topics such as the stationary phase method and Laplace’s method for curve integrals. The book integrates the study of special functions, including Euler’s Gamma and Beta functions, Bessel’s functions, and Weierstrass and Jacobi elliptic functions, with applications to nonlinear ordinary differential equations. With over 500 exercises and 150 examples, it is designed to enhance understanding and provide practice. While primarily aimed at undergraduates, the book offers valuable insights for graduate students and researchers in applied mathematics and physics. The text stands out for its treatment of the extended complex plane, orthogonal polynomials, and improper integrals, making it a versatile and advanced resource in the field. Generated by AI.

Contents:

Preface

Contents

Part I

1 Complex numbers and their properties

2 Functions of complex variable

3 Analytic functions (differentiability)

4 Integration of functions of complex variable (curve integration)

5 Cauchy theorem and Cauchy integral formulae

Exercises

Part II

6 Fundamental theorem of integration

7 Harmonic functions and mean value formulae

8 Liouville’s theorem and the fundamental theorem of algebra

9 Representation of analytic functions via the power series

10 Laurent’s expansions

11 Residues and their calculus

12 The principle of the argument and Rouche’s theorem

13 Calculation of integrals by residue theory

14 Calculation of series by residue theory

15 Entire functions

Part III

16 Conformal mappings

17 Laplace transform

18 Special functions

Bibliography

Index Generated by AI.

Notes:

Description based on publisher supplied metadata and other sources.

Part of the metadata in this record was created by AI, based on the text of the resource.

ISBN:

9783111632278

311163227X

OCLC:

1479627043