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Poisson Sum Formula

The Poisson sum formula is a special case of the general result

sum_(-infty)^inftyf(x+n)=sum_(k=-infty)^inftye^(2piikx)int_(-infty)^inftyf(x^')e^(-2piikx^')dx^'
(1)

with x=0, yielding

sum_(-infty)^inftyf(n)=sum_(k=-infty)^inftyint_(-infty)^inftyf(x^')e^(-2piikx^')dx^'.
(2)

Given phi a nonnegative, continuous, decreasing, and Riemann integrable function of [0,infty), define

psi(x)=sqrt(2/pi)int_0^inftyphi(t)cos(xt)dt.
(3)

Then the Poisson sum formula states that

sqrt(alpha)[1/2phi(0)+sum_(n=1)^inftyphi(nalpha)]=sqrt(beta)[1/2psi(0)+sum_(n=1)^inftypsi(nbeta)]
(4)

whenever alphabeta=2pi (Hardy 1999, p. 14). It follows from this formula that

sqrt(alpha)[1/2+sum_(n=1)^inftye^(-alpha^2n^2/2)]=sqrt(beta)[1/2+sum_(n=1)^inftye^(-beta^2n^2/2)]
(5)

(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.

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Poisson Sum Formula

Cite this as:

Weisstein, Eric W. "Poisson Sum Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PoissonSumFormula.html

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