A question about the Theta functions

Question

( I have seen this question: How to start with elliptic functions but it has no answer. )

In Mathematica ( this is not a Mathematica question ) I calculated the following sum:

Sum[E^(-\[Pi] n^2 x), {n, 1, Infinity}]

The answer is:

1/2 (-1 + EllipticTheta[3, 0, E^(-\[Pi] x)])

In my Mathematics studies I never came across this EllipticTheta function.

Which standard university mathematics course introduces this function?

What is your suggested book on the subject and in which standard ( graduate ? ) course is it taught?

Answer 1

EllipticTheta[1] to EllipticTheta[4] are also called Jacobi theta functions: $$ \vartheta_3(z,q) = \sum _{n=-\infty}^{\infty }{q}^{{n}^{2}}{{\rm e}^{2\,inz}} $$

So Bellman may be OK. The book on Ramanujan theta functions probably assumes you already know about Jacobi theta functions.

Probably Mathematica has documentation on its functions ... Here is the reference that Maple's documentation recommends:

Chapter 16, "Jacobian Elliptic Functions and Theta Functions" of Handbook of Mathematical Functions edited by Abramowitz and Stegun