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
[1]
![{\displaystyle \sum _{n=0}^{\infty }f\left(a+n\right)=\int _{a}^{\infty }f\left(x\right)dx+{\frac {f\left(a\right)}{2}}+\int _{0}^{\infty }{\frac {f\left(a-ix\right)-f\left(a+ix\right)}{i\left(e^{2\pi x}-1\right)}}dx}](//wikimedia.org/api/rest_v1/media/math/render/svg/f86a324e5a06638c855afdd4505d6ddd95c91ff4)
For the case
we have
![{\displaystyle \sum _{n=0}^{\infty }f(n)={\frac {1}{2}}f(0)+\int _{0}^{\infty }f(x)\,dx+i\int _{0}^{\infty }{\frac {f(it)-f(-it)}{e^{2\pi t}-1}}\,dt.}](//wikimedia.org/api/rest_v1/media/math/render/svg/19cc6618d30b7bec5700e2ea7e696c341f9006dd)
It holds for functions ƒ 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 |ƒ| 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,
![{\displaystyle \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}},}](//wikimedia.org/api/rest_v1/media/math/render/svg/a48fad2ef1a1e52d50ed8f6f0f52a50d96d31b63)
which holds for all
, s ≠ 1. Another powerful example is applying the formula to the function
: we obtain
where
is the gamma function,
is the polylogarithm and
.
Abel also gave the following variation for alternating sums:
![{\displaystyle \sum _{n=0}^{\infty }(-1)^{n}f(n)={\frac {1}{2}}f(0)+i\int _{0}^{\infty }{\frac {f(it)-f(-it)}{2\sinh(\pi t)}}\,dt,}](//wikimedia.org/api/rest_v1/media/math/render/svg/a451a24ef082f47d33ddc545bbbc4a35690b3c9c)
which is related to the Lindelöf summation formula [2]
![{\displaystyle \sum _{k=m}^{\infty }(-1)^{k}f(k)=(-1)^{m}\int _{-\infty }^{\infty }f(m-1/2+ix){\frac {dx}{2\cosh(\pi x)}}.}](//wikimedia.org/api/rest_v1/media/math/render/svg/d9785f196c7498f05b49cb0e7fa2229ee7133c77)