Poisson Sum Formula
The Poisson sum formula is a special case of the general result
| [画像: sum_(-infty)^inftyf(x+n)=sum_(k=-infty)^inftye^(2piikx)int_(-infty)^inftyf(x^')e^(-2piikx^')dx^' ] |
(1)
|
with x=0, yielding
Given phi a nonnegative, continuous, decreasing, and Riemann integrable function of [0,infty), define
Then the Poisson sum formula states that
![画像: sqrt(alpha)[1/2phi(0)+sum_(n=1)^inftyphi(nalpha)]=sqrt(beta)[1/2psi(0)+sum_(n=1)^inftypsi(nbeta)]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/PoissonSumFormula/NumberedEquation4.svg){kind=link}
whenever alphabeta=2pi (Hardy 1999, p. 14). It follows from this formula that
![画像: sqrt(alpha)[1/2+sum_(n=1)^inftye^(-alpha^2n^2/2)]=sqrt(beta)[1/2+sum_(n=1)^inftye^(-beta^2n^2/2)]](/index.cgi/contrast/https://mathworld.wolfram.com/images/equations/PoissonSumFormula/NumberedEquation5.svg){kind=link}
(Apostol 1974, pp. 322-333; Borwein and Borwein 1987, pp. 36-40).
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References
Apostol, T. M. Mathematical Analysis. Reading, MA: Addison-Wesley, 1974.Borwein, J. M. and Borwein, P. B. "Poisson Summation." §2.2 in Pi & the AGM: A Study in Analytic Number Theory and Computational Complexity. New York: Wiley, 1987.Hardy, G. H. Ramanujan: Twelve Lectures on Subjects Suggested by His Life and Work, 3rd ed. New York: Chelsea, 1999.Morse, P. M. and Feshbach, H. Methods of Theoretical Physics, Part I. New York: McGraw-Hill, pp. 466-467, 1953.Referenced on Wolfram|Alpha
Poisson Sum FormulaCite this as:
Weisstein, Eric W. "Poisson Sum Formula." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/PoissonSumFormula.html