Abel–Plana formula

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In mathematics, the Abel–Plana formula is a summation formula discovered independently by Niels Henrik Abel (1823) and Giovanni Antonio Amedeo Plana (1820). It states that

\sum_{n=0}^\infty f(n)= \int_0^\infty f(x) \, dx+ \frac 1 2 f(0)+i \int_0^\infty \frac{f(i t)-f(-i t)}{e^{2\pi t}-1} \, dt.

It holds for functions f that are holomorphic in the region Re(z) ≥ 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. (Olver 1997, p.290).

An example is provided by the Hurwitz zeta function,

\zeta(s,\alpha)= \sum_{n=0}^\infty \frac{1}{(n+\alpha)^{s}} = 
\frac{\alpha^{1-s}}{s-1} + \frac 1{2\alpha^s} + 2\int_0^\infty\frac{\sin\left(s \arctan \frac t \alpha\right)}{(\alpha^2+t^2)^\frac s 2}\frac{dt}{e^{2\pi t}-1}.

Abel also gave the following variation for alternating sums:

\sum_{n=0}^\infty (-1)^nf(n)=  \frac {1}{2} f(0)+i \int_0^\infty \frac{f(i t)-f(-i t)}{2\sinh(\pi t)} \, dt.

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