This text provides a detailed presentation of the main results for infinite products, as well as several applications. The target readership is a student familiar with the basics of real analysis of a single variable and a first course in complex analysis up to and including the calculus of residues. The book provides a detailed treatment of the main theoretical results and applications with a goal of providing the reader with a short introduction and motivation for present and future study. While the coverage does not include an exhaustive compilation of results, the reader will be armed with an understanding of infinite products within the course of more advanced studies, and, inspired by the sheer beauty of the mathematics. The book will serve as a reference for students of mathematics, physics and engineering, at the level of senior undergraduate or beginning graduate level, who want to know more about infinite products. It will also be of interest to instructors who teach courses that involve infinite products as well as mathematicians who wish to dive deeper into the subject. One could certainly design a special-topics class based on this book for undergraduates. The exercises give the reader a good opportunity to test their understanding of each section.
Author(s): Charles H. C. Little, Kee L. Teo, Bruce van Brunt
Series: Springer Undergraduate Mathematics Series
Edition: 1
Publisher: Springer Nature Switzerland AG
Year: 2022
Language: English
Pages: 251
City: Cham, Switzerland
Tags: infinite products text infinite products, Abel's Limit Theorem, Gamma function, Stirling's formula, Beta function, Jacobi triple product identity, Blaschke products, Weierstrass products, Weierstrass factorization theorem
Preface
Contents
1 Introduction
1.1 Series
1.2 Series with Non-Negative Terms
1.3 Series with General Terms
1.4 Uniform Convergence of Sequences and Series of Functions
1.5 Analytic Functions and Power Series
1.6 Double Series
2 Infinite Products
2.1 Introduction
2.2 Convergence of Products and Series
2.3 Conditionally Convergent Products
2.4 Uniform Convergence of Products of Functions
2.5 Infinite Products of Real Functions
2.6 Infinite Product Expansions for sinx and cosx
2.7 Abel's Limit Theorem for Infinite Products
2.8 Weierstrass Products
2.9 The Weierstrass Factorization Theorem
2.10 Blaschke Products
2.11 Double Infinite Products
3 The Gamma Function
3.1 Representations of the Gamma Function
3.2 Some Identities Involving the Gamma Function
3.3 Analytic Functions Related to
3.4 Stirling's Formula
3.5 Applications to Products and Series
3.6 The Beta Function
4 Prime Numbers, Partitions and Products
4.1 Prime Numbers and Euler's Identity
4.2 Partition Functions
4.3 The Jacobi Triple Product Identity
5 Epilogue
5.1 Product Representation of Functions
5.2 Infinite Products for Constants
6 Tables of Products
Bibliography
Index
