holomorphic continuation

Question

consider the function given by $f(t):=\sum\limits_{n=0}^{\infty}e^{-\left(n+\frac{1}{2}\right)^2t}$ for $t\in (0,\infty)$.

This function can be continued holomorphically for all complex numbers with positive real part $\Re(z)>0$ by the same formula.

My Question is: How can I prove that there must be an holomorphic continuation to the whole plane? Are there any (simple) Poles for the continuation?

Any help will be very appreciated!

Best regards,

Answer 1

On the contrary: the Jacobi theta function $$\theta_2(0,q) = 2 q^{1/4}\sum_{n=0}^\infty q^{n(n+1)}$$ has a natural boundary at $|q|=1$. Your function is $f(t) = (1/2) \theta_2(0,\exp(-t))$, so you can't continue into the left half-plane.