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
(1)
where
are the zeros of
and
are the poles contained within the contour. An appropriate choice of and then yields
(2)
or equivalently
(3)
The formula is particularly useful in Casimir effect calculations involving differences between quantized modes and free modes.
are the zeros of 
and 
are the poles contained within the contour 
. An appropriate choice of 
and 
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,](/images/equations/Abel-PlanaFormula/NumberedEquation2.svg)
