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 $\theta_{3}(q)$.

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.

Two Approaches to Trigonometry

Trigonometric Functions

Trigonometry is the study of the relationships between angles and sides of a triangle. This is the way it is introduced in secondary classes. The basic idea here is to use the concept of "similarity" of two triangles in a slightly formalized way and use it for practical applications like "heights and distances".

Two Problems from IIT-JEE

IIT-JEE (Joint Entrance Examination for admission to Indian Institutes of Technology) is the most prestigious examination at 10+2 level in India. The questions are supposed to be tough and require ingenuity on the part of the student. There is a booming ecosystem of books, coaching classes and correspondence classes around IIT-JEE. But I find this ecosystem to be quite detrimental in improving the abilities of the student.

Gauss and Regular Polygons: Conclusion

The Central Result

This is the concluding post in this series and we aim to prove the following result (proved in part by Gauss and finally the converse by Wantzel):
A regular polygon of $n, n > 2$ sides can be constructed by Euclidean tools if and only if $\phi(n) = 2^{k}$.

Gauss and Regular Polygons: Gaussian Periods Contd.

Properties of Gaussian Periods

In this post we are going to establish the following properties of the Gaussian periods which will ultimately lead to a solution of the equation $z^{p} - 1 = 0$. Again as in previous post, $p$ is to be considered a prime unless otherwise stated. In the following we have $e, f$ as two positive integers with $ef = (p - 1)$.
1. Any period of $f$ terms can be expressed as a polynomial in any other period of $f$ terms with rational coefficients.
2. If $g$ divides $(p - 1)$ and $f$ divides $g$, then any period of $f$ terms is a root of a polynomial equation of degree $g / f$ whose coefficients are rational expressions of a period of $g$ terms.