Abel-Plana Formula (original) (raw)

The Abel-Plana formula gives an expression for the difference between a discrete sum and the corresponding integral. The formula can be derived from the argument principle

∮_gammaf(z)(g^'(z))/(g(z))dz=sum_(n)f(mu_n)-sum_(m)f(nu_m), (1)

where mu_n are the zeros of g(z) and nu_m are the poles contained within the contour gamma. An appropriate choice of g and gamma then yields

sum_(n=0)^inftyf(n)-int_0^inftyf(x)dx=1/2f(0)-1/2int_0^infty[f(it)-f(-it)][cot(piit)+i]dt, (2)

or equivalently

sum_(n=0)^inftyf(n)-int_0^inftyf(x)dx=1/2f(0)+iint_0^infty(f(it)-f(-it))/(e^(2pit)-1)dt. (3)

The formula is particularly useful in Casimir effect calculations involving differences between quantized modes and free modes.

See also

Argument Principle

This entry contributed by David Anderson

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References

Mostepanenko, V. M. and Trunov, N. N. §2.2 in The Casimir Effect and Its Applications. Oxford, England: Clarendon Press, 1997.Saharian, A. A. "The Generalized Abel-Plana Formula. Applications to Bessel Functions and Casimir Effect." http://www.ictp.trieste.it/~pub_off/preprints-sources/2000/IC2000014P.pdf.

Referenced on Wolfram|Alpha

Abel-Plana Formula

Cite this as:

Anderson, David. "Abel-Plana Formula." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Abel-PlanaFormula.html

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