Abel–Plana formula


In mathematics, the Abel–Plana formula is a summation formula discovered independently by and. It states that
It holds for functions f that are holomorphic in the region Re ≥ 0, and satisfy a suitable growth condition in this region; for example it is enough to assume that |f| is bounded by C/|z|1+ε in this region for some constants C, ε > 0, though the formula also holds under much weaker bounds..
An example is provided by the Hurwitz zeta function,
which holds for all,.
Abel also gave the following variation for alternating sums:

Proof

Let be holomorphic on, such that, and for,. Taking with the residue theorem
Then
Using the Cauchy integral theorem for the last one., thus obtaining
This identity stays true by analytic continuation everywhere the integral converges, letting we obtain Abel-Plana's formula
The case f ≠ 0 is obtained similarly, replacing by two integrals following the same curves with a small indentation on the left and right of 0.