Intro
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 ( )
(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− 2
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 √ ( )
7.2 Compactification and Elliptic Curves
7.2.1 Embedding of R into the projective plane
7.2.2 Another way. (Embedding into (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 (Jacobi's Theorem)