Preface
Acknowledgements
Contents
Chapter 0 Introduction
0.1 What is an Elliptic Function in one Phrase?
0.2 What Properties Do Elliptic Functions Have?
0.3 What Use Are Elliptic Functions?
Pendulum
Skipping rope
Soliton equations
Solvable lattice models
Arithmetic-geometric mean
Formula for solving quintic equations
0.4 A Small Digression on Elliptic Curves
0.5 Structure of this book
Part I Real Part
Chapter 1 The Arc Length of Curves
1.1 The Arc Length of the Ellipse
1.2 The Lemniscate and its Arc Length
Chapter 2 Classification of Elliptic Integrals
2.1 What is an Elliptic Integral?
2.2 Classification of Elliptic Integrals
(I) Standardising 𝝋(𝒙)
(II) Standardising the elliptic integral
2.3 Real Elliptic Integrals.
Chapter 3 Applications of Elliptic Integrals
3.1 The Arithmetic-Geometric Mean
3.2 Motion of a Simple Pendulum
Chapter 4 Jacobi’s Elliptic Functions on
4.1 Jacobi’s Elliptic Functions
4.2 Properties of Jacobi’s Elliptic Functions
4.2.1 Troika of Jacobi’s elliptic functions
4.2.2 Derivatives
4.2.3 Addition formulae
Chapter 5 Applications of Jacobi’s Elliptic Functions
5.1 Motion of a Simple Pendulum
5.2 The Shape of a Skipping Rope
5.2.1 Derivation of the differential equation
5.2.2 Solution of the differential equation and an elliptic function
5.2.3 The variational method
Part II Complex Part
Chapter 6 Riemann Surfaces of Algebraic Functions
6.1 Riemann Surfaces of Algebraic Functions
6.1.1 What is the problem?
6.1.2 Then, what should we do?
6.1.3 Another construction
6.1.4 The Riemann surface of √1− 𝒛2
6.2 Analysis on Riemann Surfaces
6.2.1 Integrals on Riemann surfaces
6.2.2 Homology groups (a very short crash course)
6.2.3 Periods of one-forms
Chapter 7 Elliptic Curves
7.1 The Riemann Surface of √𝝋(𝒛)
7.2 Compactification and Elliptic Curves
7.2.1 Embedding of R into the projective plane
7.2.2 Another way. (Embedding into 𝑶(2))
7.3 The Shape of R
Chapter 8 Complex Elliptic Integrals
8.1 Complex Elliptic Integrals of the First Kind
8.2 Complex Elliptic Integrals of the Second Kind
8.3 Complex Elliptic Integrals of the Third Kind
Chapter 9 Mapping the Upper Half Plane to a Rectangle
9.1 The Riemann Mapping Theorem
9.2 The Reflection Principle
9.3 Holomorphic Mapping from the Upper Half Plane to a Rectangle
9.4 Elliptic Integrals on an Elliptic Curve
Chapter 10 The Abel–Jacobi Theorem
10.1 Statement of the Abel–Jacobi Theorem
10.2 Abelian Differentials and Meromorphic Functions on an Elliptic Curve
10.2.1 Abelian differentials of the first kind
10.2.2 Abelian differentials of the second/third kinds and meromorphic functions
10.2.3 Construction of special meromorphic functions and Abelian differentials
10.3 Surjectivity of 𝑨𝑱 (Jacobi’s Theorem)
10.3.1 Finding an inverse image
10.3.2 Topological proof of surjectivity of 𝑨𝑱
10.4 Injectivity of 𝑨𝑱 (Abel’s Theorem)
Chapter 11 The General Theory of Elliptic Functions
11.1 Definition of Elliptic Functions
11.2 General Properties of Elliptic Functions
Chapter 12 TheWeierstrass ℘-Function
12.1 Construction of the ℘-Function
12.2 Properties of ℘(𝒖)
12.3 Weierstrass Zeta and Sigma Functions
12.4 Addition Theorems for the ℘-Function.
Chapter 13 Addition Theorems
13.1 Addition Theorems of the ℘-Function Revisited
13.2 Addition Formulae of General Elliptic Functions
13.3 Resultants
Chapter 14 Characterisation by Addition Formulae
14.1 Behaviour of Meromorphic Functions at Infinity
14.2 Properties of Essential Singularities
14.3 Proof of the Weierstrass–Phragmén Theorem
Chapter 15 Theta Functions
15.1 Definition of Theta Functions
15.2 Properties of Theta Functions
15.2.1 Quasi-periodicity
15.2.2 Parity
15.2.3 Heat equations
15.2.4 Zeros
15.3 Jacobi’s Theta Relations
15.4 Jacobi’s Derivative Formula
15.5 Modular Transformations of Theta Functions
Chapter 16 Infinite Product Factorisation of Theta Functions
16.1 Infinite Product of Functions
16.2 Infinite Product Factorisation of Theta Functions
Chapter 17 Complex Jacobian Elliptic Functions
17.1 Definition of Jacobi’s Elliptic Functions as Complex Functions
17.2 Inversion from 𝒌2 to 𝝉
17.3 Properties of sn(𝒖, 𝒌)
17.3.1 The addition theorem of sn
17.3.2 Limit as 𝒌 → 0
17.3.3 Jacobi’s imaginary transformation and the limit as 𝒌 → 1
17.3.4 The periods of sn are 4𝑲(𝒌) and 2𝒊𝑲′ (𝒌)
17.4 The Arithmetic-Geometric Mean Revisited
Appendix A Theorems in Analysis and Complex Analysis
A.1 Convergence Theorems of Integrals
A.2 Several Facts in Complex Analysis
A.2.1 Integral depending on a parameter
A.2.2 Consequences of the Taylor expansion
A.2.3 Weierstrass’s double series theorem
A.2.4 The argument principle and its generalisation
References
Index