How does one express Jacobi theta functions in terms of Elliptic Integrals?

Question

It is known that elliptic functions may be expressed in terms of Jacobi theta functions. Moreover, by construction, the elliptic integrals are inverses of elliptic functions. It therefore seems to be that one may express each Jacobi theta function $\theta_k(z\tau)$, $k=1,2,3,4$, e.g. $\theta_3(z;\tau)=\sum_{n=-\infty}^{n=\infty}\exp\left( 2\pi i n z +\pi i \tau n^2 \right)$, in terms of elliptic integrals!

There are many references on these topics with various types of notation but I am finding it hard to reconcile these sources and solve this problem. Is this problem solved somewhere? Any solution or further reference would be appreciated!

Answer 1

Theta functions are intimately related to elliptic integrals and elliptic functions. In particular their values at $z=0$ which also go by the name thetanulls have a direct relationship with elliptic integrals.

Here is a brief summary of such key relationships. Let's start with a number $k\in(0,1)$ which is called elliptic modulus and let $k'=\sqrt{1-k^2}$ be the complementary (to $k$) modulus. We then define complete elliptic integral of first kind $$K(k) =\int_{0}^{\pi/2}\frac{dx}{\sqrt{1-k^2\sin^2x}}\tag{1}$$ and complete elliptic integral of second kind $$E(k) =\int_{0}^{\pi/2}\sqrt{1-k^2\sin^2x}\,dx\tag{2}$$ If the values of $k, k'$ are available from context then $K(k), K(k'), E(k), E(k') $ are usually denoted by $K, K', E, E'$ and they satisfy the fundamental identity $$KE'+K'E-KK'=\frac{\pi} {2}\tag{3}$$ The theta functions play a role in inverting these functions. Thus if the values of $K, K'$ are known then the values of $k, k'$ can be obtained as functions of a parameter $q$ defined by $$q=\exp\left(-\pi\frac{K(k')} {K(k)} \right) =e^{-\pi K'/K} \tag{4}$$ which is also called nome corresponding to modulus $k$. The nome $q$ is also related to the parameter $\tau$ used in definition of theta functions in your question via $$q=e^{\pi i\tau} ,\tau=i\frac{K'}{K}\tag{5}$$ and we have the following formulas $$k=\frac{\vartheta_{2}^{2}(q)} {\vartheta_{3}^{2}(q)},k'=\frac{\vartheta_{4}^{2}(q)} {\vartheta_{3}^{2}(q)},K=\frac{\pi}{2}\vartheta_{3}^{2}(q), \vartheta_{i} (q) =\vartheta_{i} (0;\tau)\tag{6}$$ These formulas also help you evaluate thetanulls in terms of elliptic integrals and moduli.

Here is deep and important result which is key to certain closed form evaluations:

Theorem (due to Jacobi, Abel and Ramanujan): If the ratio $K'/K$ is the square root of a positive rational number $n$, then the corresponding modulus $k_n$ is an algebraic number. Such moduli are famous by the name singular moduli.

A number of mathematicians (most famously Ramanujan) found closed form expressions for modulus $k_n$ corresponding to many positive integers $n$. Next comes the surprising result:

Theorem (due to Selberg and Chowla): If $k$ is a singular modulus then $K, E$ can be evaluated in closed form in terms of $\pi$ and values of Gamma function at rational points.

Using these theorems you can thus evaluate the values of theta functions for $z=0,\tau=i\sqrt{n}$ where $n$ is a positive rational number.