Write $\sum_{n=1}^{\infty}e^{-\frac{\beta k}{2}(\frac{n\pi}{l})^{2}}\cos(\frac{n\pi z}{l})$ as a sum of Gaussians

Question

I'm studying a paper (Numerical simulation of Gaussian chains near hard surfaces) and in an appendix they show that $$\sum_{n=1}^{\infty}e^{-\frac{\beta k}{2}(\frac{n\pi}{l})^{2}}\cos(\frac{n\pi z}{l})$$ can be written as an infinite sum of Gaussians using Poisson summation, but I cannot really follow the precise steps for that. Any help would be appreciated. Also, would the same formula be valid replacing $z$ by $z+\pi/2$ or anything like that? I guess so but just double checking (my Fourier game is weak)

Answer 1

By writing $\cos u = \frac 1 2 \left( e^{iu}+e^{-iu}\right)$, $$\begin{split} \sum_{n=1}^{+\infty}e^{-\frac{\beta k}{2}(\frac{n\pi}{l})^{2}}\cos\left(\frac{n\pi z}{l}\right)&=-\frac 1 2+\frac 1 2 \cdot\sum_{n=-\infty}^{+\infty}e^{-\frac{\beta k}{2}(\frac{n\pi}{l})^{2}}e^{\frac{in\pi z}{l}}\\ &= -\frac 1 2+\frac 1 2 \cdot\sum_{n=-\infty}^{+\infty}f(n) \end{split}$$ where $$f(x)=e^{-\frac{\beta k}{2}\left ( \frac {\pi x}l\right)^2 } e^{\frac{i\pi xz}{l}}$$ By Poisson's summation formula $$\sum_{n=-\infty}^{+\infty}f(n)=\sum_{p=-\infty}^{+\infty}\widehat f(p)$$ So all you need is to compute the Fourier transform of $f$. If I'm not mistaken, $$\widehat f(\xi) = \sqrt{\frac{2}{\pi\beta k}}le^{-{\frac{\left(2\xi-\frac z l\right)^2}{2\beta k}}}$$

Can you finish?