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

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,

which holds for all s, s ≠ 1.

Abel also gave the following variation for alternating sums:


Let be holomorphic on , such that , and for , . Taking with the residue theorem


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) ≠ 0 is obtained similarly, replacing by two integrals following the same curves with a small indentation on the left and right of 0.

See also[edit]


  • 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, S2CID 54634413
  • Olver, Frank William John (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

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