# Jacobi Eliiptic Theta Function

1. Feb 25, 2008

### Pere Callahan

Hi,

I bother you again witha question concerning special functions.

I have to evaluate

$$\prod_{k=1}^\infty{(1+\frac{x^2}{{4^k}})}$$

From Mathworld ( http://mathworld.wolfram.com/InfiniteProduct.html formula 59) I know, that for x=1 this can be written as

$$4^{\frac{1}{24}}\left(\theta_4\left[0,\frac{1}{4}\right]\right)^{-\frac{1}{2}}\left(\frac{1}{2} \theta_1'\left[0,\frac{1}{4}\right]\right)^{\frac{1}{6}}$$

where $$\theta_n\left[z,q\right]$$ are the "famous" Elliptic Theta functions of Jacobi.

Is there any chance to extend this to arbitrary real x?

Mathworld and Abramowitz seem not to help me any further.

Thanks
-Pere

A related representation ist the logarithmic derivative which I found to be

$$2x\sum_{k=1}^\infty{\frac{1}{4^k+x^2}}$$

Any ideas what function this might be ... for x=0 it's easy at least ..

Last edited: Feb 25, 2008