Elliptic Functions: A Constructive Approach

229 Pages·1996·7.988 MB·English

by  Peter L. Walker

Elliptic Functions

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ISBN O 471 96531 6
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Contents
Preface ..... . Xlll
0 Preliminaries . 1
0 .1 Differentiation 1
0.2 Connected Sets 2
0.3 Sequences and Series 3
0 .4 Sequences of Functions . 5
0.5 Infinite Products .. . 7
0.6 Integration ..... . 8
0.7 Exercises for Chapter 0 10
1 Circular Functions 11
1.1 Eisenstein Series 13
1.2 Addition Formulae 15
1.3 Infinite Products 18
1.4 An Alternating Series 19
1.5 The Exponential Function 20
1.6 Power Series ....... . 22
1.7 Logarithms ....... . 23
1.8 Inverse Circular Functions 27
1. 9 Geometry . . . . . . 31
1.10 Asymptotic Estimates 35
1.11 Some Alternatives . . 36
1.12 Exercises for Chapter 1 39
2 Gamma and Related Functions 43
2 .1 Euler Series . . . . . . . . . . 43
2.2 The Gamma Function as an Infinite Product 46
2.3 The Gamma Function as an Integral 49
2.4 Beta Integrals ..... 52
2.5 Asymptotic Estimates ...... . 54
viii CONTENTS
2.6 Fourier Transforms . . . 62
2.7 Exercises for Chapter 2 66
3 Basic Elliptic Functions 69
3.1 Lattices ........ 70
3.2 Basic Elliptic Functions 71
3.3 Addition Formulae ... 77
3.4 Weierstrass Functions 80
3.5 Exercises for Chapter 3 84
4 Theta Functions . . . . . 87
4.1 Phi and its Translates 87
4.2 Theta Series . . . . . . 91
4.3 Sums of Squares ... 93
4.4 Addition and Rotation Formulae 95
4.5 Fourier Series ...... 98
4.6 Exercises for Chapter 4 102
5 Jacobian Functions . 105
5.1 Alternating Series 106
5.2 Addition Formulae 112
5.3 Derivatives and Sums of Squares 114
5.4 Fourier Series ...... 115
5.5 Exercises for Chapter 5 118
6 Elliptic Integrals .... 121
6.1 The Simple Pendulum 122
6.2 Glaisher's Notation .. 126
6.3 Elliptic Integrals ... 131
6.4 The Length of an Ellipse . 136
6.5 The Amplitude and the Modular Angle 138
6.6 The Arithmetic-geometric Mean 141
6.7 Exercises for Chapter 6 146
7 Modular Functions ... 151
7.1 Modular Groups ... 151
7.2 Fundamental Regions 153
7.3 The Elliptic Modular Function 156
7.4 Other Modular Functions 161
7.5 Summary and Special Cases 164
7.6 Exercises for Chapter 7 168
8 Applications 171
8.1 Waves .. 171
CONTENTS ix
8.2 Entire Functions 177
8.3 Some Number Theory 180
8.4 Elliptic Curves .... 188
8.4.1 Elliptic Curves over C 188
8.4.2 Elliptic Curves over R 191
8.4.3 Elliptic Curves over Q 195
8.4.4 Finite fields ... 203
8.5 Exercises for Chapter 8 207
References . 211
Index .... 213