Welcome everybody :)
I need your help in answering the following question:
Let $t > 0$ and $\mathrm{G}_t(x) = (2\pi t)^{-1/2}e^{-x^2/2t}$
Show that $$\mathrm{P}(t,x) = \sum_{n\in\mathbb{Z}} \mathrm{G}_t(x-2\pi n)$$
is a smooth function on $(0,\infty)\times\mathbb{R}$, i.e. an element of $\mathbb{C}^\infty((0,\infty)\times\mathbb{R})$.
Somebody has told me that this $\mathrm{G}_t$ is called the Gauss-Weierstrass kernel.
I have two approaches to show the task:
a) use the Weierstrass M-Test
b) use the Poisson summation formula
However I'm not sure if there's another approach using complex analysis (Morera's Theorem).
My aim here is to ask some tough guys about their opinion how to solve this task and more preferable posting a solution if possible.
Thank you in advance :)