Poisson Sum Formula

A special case of the general result

$$\sum_{n=-\infty}^\infty f(x+n)=\sum_{k=-\infty}^\infty e^{2\pi ikx}\int_{-\infty}^\infty f(x_1)e^{-2\pi ikx}\,dx_1$$ (1)

with $x=0$, yielding

$$\sum_{n=-\infty}^\infty f(n)=\sum_{k=-\infty}^\infty \int_{-\infty}^\infty f(x_1)e^{-2\pi ikx}\,dx_1.$$ (2)

An alternate form is

$${1\over 2}+\sum_{n=1}^\infty e^{-(nx)^2} = {\sqrt{\pi}\over x}\left[{{1\over 2}+\sum_{n=1}^\infty e^{-(n\pi/x)^2}}\right].$$ (3)

Another formula called the Poisson summation formula is

$$\sqrt{\alpha}\, [{\textstyle{1\over 2}}\phi(0)+\phi(\alpha)+... ...\textstyle{1\over 2}}\psi(0)+\psi(\beta)+\psi(2\beta)+\ldots],$$ (4)

where

$$\psi(x) = \sqrt{2\over\pi} \int_0^\infty \psi(t)\cos(xt)\,dt$$ (5)

$$\alpha\beta = 2\pi.$$ (6)

References

Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 466-467, 1953.