Elliptic Functions: Introduction

Introduction

In the previous posts we have covered introductory material on the following topics like elliptic integrals, AGM, and theta functions. All the concepts are tightly coupled with each other and belong more properly to the theory of elliptic functions. The theory of elliptic functions puts all the above concepts into a unified perspective and provides us a coherent picture. The approach to elliptic functions would be again very introductory and we will not pursue the topics related to "theory of functions of complex variable" in detail.

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The Magic of Theta Functions: Contd.

In the previous post we studied some interesting properties of theta functions which were used to relate them to AGM and thereby to elliptic integrals. We will continue to explore further in this direction and start with a remarkable property of theta function $ \theta_{3}(q)$.

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The Magic of Theta Functions

Introduction

Theta functions were originally introduced by Carl Gustav Jacob Jacobi while studying elliptic functions (which are in turn related to elliptic integrals). These functions are also connected with number theory and they have many interesting properties besides. Since they are related to elliptic integrals and we have seen in a previous post that the elliptic integrals are related to the AGM (arithmetic-geometric mean), it follows that the theta functions are related to the AGM. We will cover these topics in this series of posts and will also mention some number theoretic applications of theta functions.

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