Abel-Plana Formula

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


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


or equivalently


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

Explore with Wolfram|Alpha


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."

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.

Subject classifications