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,



holomorphic continuation
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.


