Theta functions, elliptic functions and ?

by Heng Huat Chan

Overview

This book presents several results on elliptic functions and Pi, using Jacobi’s triple product identity as a tool to show suprising connections between different topics within number theory such as theta functions, Eisenstein series, the Dedekind delta function, and Ramanujan’s work on Pi. The included exercises make it ideal for both classroom use and self-study.

Product Details

ISBN-13:	9783110540710
Publisher:	De Gruyter
Publication date:	07/06/2020
Series:	De Gruyter Textbook
Pages:	138
Product dimensions:	6.10(w) x 9.06(h) x 0.00(d)
Age Range:	18 Years

About the Author

Heng Huat Chan, National University of Singapore, Singapore.

Table of Contents

Foreword IX

Introduction XI

Acknowledgments XV

1 An introduction to Jacobi's triple product identity 1

1.1 Cauchy's generalization of the binomial theorem 1

1.2 Jacobi's triple product identity 4

1.3 Two identities of Euler 6

1.4 Andrews' proof of Jacobi's triple product identity 9

1.5 Jacobi's triple product identity and the proof of the quintuple product identity 10

2 Jacobi's theta functions of one variable and the triple product identity 15

2.1 Jacobi's theta functions of one variable 15

2.2 Identities associated with Jacobi's theta functions of one variable 16

2.3 Jacobi's triple product identity revisited 18

2.4 A Fermat-type equation and Jacobi's theta functions 21

2.5 A quotient of Jacobi's theta functions 23

3 Two-variable extensions of Jacobi's theta functions and the partition function 29

3.1 Two-variable extensions of Jacobi's theta functions 29

3.2 An important identity of Jacobi 30

3.3 The partition function p(n) and its generating function 31

3.4 The values of p(n) 33

3.5 Ramanujan's congruences for p(n) 35

4 Ramanujan's differential equations 41

4.1 The Bernoulli numbers and cot u 41

4.2 The logarithmic derivative of $$$₁(υ/τ) 42

4.3 Ramanujan's differential equations for L₂(x), L₄(x) and L₆(x) 44

5 Elliptic functions and Jacobi's triple product identity 53

5.1 Elliptic functions 53

5.2 Transformation formulas associated with Jacobi's theta functions 54

5.3 Proof of a fundamental identity of Jacobi associated with $$$(0|π) 55

5.4 Proof of the Jacob! triple product identity using theory of elliptic functions 59

6 Two elliptic functions and their properties 63

6.1 Weierstrass' $$$ function 63

6.2 A differential equation satisfied by Weierstrass' $$$ function 64

6.3 An elliptic function $$$(u|τ) associated with $$$(u|τ) 65

6.4 Weierstrass' $$$ functions and an elliptic function associated with $$$(u|τ) 66

6.5 Transformation formula satisfied by Dedekind's function Δ(τ) 67

7 An elliptic function of Jacobi 75

7.1 An elliptic function of Jacobi 75

7.2 A differential equation satisfied by S(u|τ) 76

7.3 The first few terms in the series expansion of S(u|τ) 80

7.4 Representations of Eisenstein series E₄(τ) and E₆(τ) in terms of $$$(0|τ)and λ 81

8 Hypergeometric series and Ramanujan's series 1/π 87

8.1 Ramanujan's differential equations and Lagrange's four-square theorem 87

8.2 Hypergeometric series and a differential equation satisfied by $$$(0|τ) and λ 90

8.3 Hypergeometric series and π 96

8.4 Ramanujan's series for 1/π 97

9 The Gauss-Brent-Salamin algorithm for π 103

9.1 Gauss' arithmetic-geometric mean 103

9.2 Gauss' AGM, Jacobi's theta functions and hypergeometric series 105

9.3 The Gauss-Brent-Salamin algorithm for π 108

Index 117

Bibliography 119