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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:

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

Euler–Maclaurin summation formula

References

Abel, N.H. (1823), Solution de quelques problèmes à l’aide d’intégrales définies
Butzer, P. L.; Ferreira, P. J. S. G.; Schmeisser, G.; Stens, R. L. (2011), "The summation formulae of Euler-Maclaurin, Abel-Plana, Poisson, and their interconnections with the approximate sampling formula of signal analysis", Results in Mathematics 59 (3): 359–400, doi:10.1007/s00025-010-0083-8, ISSN 1422-6383, MR 2793463
Olver, Frank W. J. (1997) [1974], Asymptotics and special functions, AKP Classics, Wellesley, MA: A K Peters Ltd., ISBN 978-1-56881-069-0, MR 1429619
Plana, G.A.A. (1820), "Sur une nouvelle expression analytique des nombres Bernoulliens, propre à exprimer en termes finis la formule générale pour la sommation des suites", Mem. Accad. Sci. Torino 25: 403–418