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 ≠ 1.

Abel also gave the following variation for alternating sums:

Proof[edit]

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) ≠ 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]

References[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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