Elliptic Functions: Fourier Series

We have discussed various interesting properties of elliptic functions and related theta functions in detail in previous posts. In particular we discussed that these elliptic functions are doubly periodic. It is only natural then to seek the Fourier series corresponding to these functions. However in this case we will use only the real periods to expand these functions in a Fourier series. It turns out that the Fourier expansions provide us many important identities which can be used in surprisingly many ways to connect to number theory.

Elliptic Functions: Theta Function Identities

In this post we will prove some theta function identities. We will try to pass from an existing identity between the elliptic functions to a corresponding identity between theta functions. Sometimes we will also establish identities which follow quite obviously from the series or product expansions of the theta functions. Most of the times we will also make use of the Liouville's theorem that any doubly periodic entire function is a constant.

Elliptic Functions: Theta Functions Contd.

The values of theta functions at the point $z = 0$ are themselves very important and surprisingly have deep connections with number theory. For example consider $$\theta_{3}(q) = \sum_{n = -\infty}^{\infty} q^{n^{2}}$$ Then clearly $$\theta_{3}^{2}(q) = \sum_{i, j = -\infty}^{\infty} q^{i^{2} + j^{2}} = \sum_{n = 0}^{\infty} r(n)q^{n}$$ where $r(n) = r_{2}(n)$ represents the number of ways in which integer $n$ can be expressed as the sum of two squares (here we count order as well as sign separately).

Elliptic Functions: Genesis of Theta Functions

We have seen in the last post that the elliptic functions can be expressed in the form of infinite product and these products look actually like ratio of two infinite products. We wish to consider these products (which are more commonly known as theta functions) in more detail in this post.