# Poisson summation formula

In this note we will follow the presentation given by Ziman .

Consider $$f(x)$$ an arbitrary function defined in the interval $$[n,n+1]$$. We can write

$$f(x)=\sum_{s=-\infty}^{+\infty}e^{-i2\pi xs}g_s\label{fourier}$$

with $$g_s$$ given by

$$g_s=\int_n^{n+1} dx f(x) e^{i2\pi sx} \label{fourier-coefficient}\;.$$

We can evaluate Eq. \eqref{fourier} at the point $$n+\gamma$$ with $$\gamma\in[0,1]$$ to obtain

$$f(n+\gamma)=\sum_{s=-\infty}^{+\infty} e^{i2\pi s\gamma} \int_n^{n+1} dx f(x) e^{i2\pi sx}\;.\label{fn}$$

Now we can sum over $$n$$ to obtain the Poisson summation formula

$$\sum_{n=-\infty}^{+\infty}f(n+\gamma)=\sum_{s=-\infty}^{+\infty} e^{i2\pi s\gamma} \int_{-\infty}^{+\infty} dx f(x) e^{i2\pi sx}\;.\label{pois}$$

## Application: A very useful identity in Condensed Matter Physics

If $$f(x)=\delta(x)$$ then
$$\sum_{n=-\infty}^{+\infty}\delta(n+\gamma)=\sum_{s=-\infty}^{+\infty} e^{i2\pi s\gamma} \;.\label{conmat}$$

## Application: Jacobi imaginary transformation

We can use the Poisson summation formula in Eq. \eqref{pois} just derived to obtain the formula used in the evaluation of Ewald sums.

$\sum_{k=-\infty}^{+\infty} e^{-k^2t}=\sqrt{\pi \over t}\sum_{p=-\infty}^{+\infty} e^{-{\pi^2 p^2\over t}}\label{jacobi}$