Q:

Show that $\sum_{-\infty}^{\infty} (-1)^n\exp(nz-\frac{1}{2}(n+\frac{1}{2})^2\omega)$ converges and is entire

Let $f(z)=\sum_{n=-\infty}^{\infty} (-1)^n\exp(nz-\frac{1}{2}(n+\frac{1}{2})^2\omega)$ where $\Re(\omega)>0$. Show that $f$ converges and is entire.

I am not sure how to approach this problem. Do I find a function $h$ so $f$ is the Laurent series of $h$? How do I show the series converges?

A:

Hint: Estimate the absolute value of the summand, to show the series converges absolutely, uniformly on compact sets.

