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Analyticity (regularity) of an infinite series arising with the heat equation
I need a little help with this exercise.
Let $G_t(x) = (2 \pi t)^{\frac{-1}{2}} e^{\frac{-x^2}{2t}}$ and $P(t,x) = \sum_{n \in \mathbb Z} G_t(x-2 \pi n) , t > 0$ Now show: $P(t,x)$ is a smooth function on $(0,\infty)\times \mathbb R$
A hint I've seen was to add $r e^{i \theta}$ to the exponent of $G_t$ , i.e. so that you have $G_t(x + r e^{i \theta} - 2 \pi n)$ which allows you to use complex analysis. My initial intention was to show that the sum converges uniformly towards a function and get a formula for all the derivatives but I think that's too explicit for this exercise here.
Can someone help me how I go on instead? I think I need to show that $P(t,x)$ is holomorphic on all of $\mathbb C$ and then in the end let $r \rightarrow 0$ so that I have a real valued function again. I'm really not sure how to do anything in this exercise so help would be greatly appreciated.
Cheers and thx in advance!