Closed-Form Solution for f(A,B) Equation

[tom.stoer]

Does anybody know a closed for

$$f(A,B) = \sum_{n\in\mathbb{Z}}e^{-Bn^2 + iAn}$$

with A real and B positive.

 

[mfb]

This is a harder version of this related question, and we did not find an answer there.

 

[tom.stoer]

I can't believe that.
I tried to calculate the density operator in position rep. for a particle on a circle. You get the exp(iAn) term from the eigenfunctions on the circle; A contains the angle. And you get the exp(-Bn²) from exp(-H/T).
So it's nothing exotic.

 

[tom.stoer]

It IS exotic, you need the elliptic theta functions

http://m.wolframalpha.com/input/?i=Sum e^[-B*n^2+i*A*n] n=-oo to +oo&x=0&y=0