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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
\end{displaymath} (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.
\end{displaymath} (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].
\end{displaymath} (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],
\end{displaymath} (4)

$\displaystyle \psi(x)$ $\textstyle =$ $\displaystyle \sqrt{2\over\pi} \int_0^\infty \psi(t)\cos(xt)\,dt$ (5)
$\displaystyle \alpha\beta$ $\textstyle =$ $\displaystyle 2\pi.$ (6)


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

© 1996-9 Eric W. Weisstein