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]
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.}](https://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}},}](https://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,}](https://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)}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9785f196c7498f05b49cb0e7fa2229ee7133c77)
Proof
Let
be holomorphic on
, such that
,
and for
,
. Taking
with the residue theorem
![{\displaystyle \int _{a^{-1}\infty }^{0}+\int _{0}^{a\infty }{\frac {f(z)}{e^{-2i\pi z}-1}}\,dz=-2i\pi \sum _{n=0}^{\infty }\operatorname {Res} \left({\frac {f(z)}{e^{-2i\pi z}-1}}\right)=\sum _{n=0}^{\infty }f(n).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d0d520713a049a92b25eb77c312f8d852272589a)
Then
![{\displaystyle {\begin{aligned}\int _{a^{-1}\infty }^{0}{\frac {f(z)}{e^{-2i\pi z}-1}}\,dz&=-\int _{0}^{a^{-1}\infty }{\frac {f(z)}{e^{-2i\pi z}-1}}\,dz\\&=\int _{0}^{a^{-1}\infty }{\frac {f(z)}{e^{2i\pi z}-1}}\,dz+\int _{0}^{a^{-1}\infty }f(z)\,dz\\&=\int _{0}^{\infty }{\frac {f(a^{-1}t)}{e^{2i\pi a^{-1}t}-1}}\,d(a^{-1}t)+\int _{0}^{\infty }f(t)\,dt.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/650fc26c23bd1fb2c460730ca0531f56357973fe)
Using the Cauchy integral theorem for the last one.
![{\displaystyle \int _{0}^{a\infty }{\frac {f(z)}{e^{-2i\pi z}-1}}\,dz=\int _{0}^{\infty }{\frac {f(at)}{e^{-2i\pi at}-1}}\,d(at),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0126f20d441ce5eec206d513dacbad968e0f06db)
thus obtaining
![{\displaystyle \sum _{n=0}^{\infty }f(n)=\int _{0}^{\infty }\left(f(t)+{\frac {a\,f(at)}{e^{-2i\pi at}-1}}+{\frac {a^{-1}f(a^{-1}t)}{e^{2i\pi a^{-1}t}-1}}\right)\,dt.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/07b8cbefe6abae53ababab46996456086eceedd1)
This identity stays true by analytic continuation everywhere the integral converges, letting
we obtain the Abel–Plana formula
![{\displaystyle \sum _{n=0}^{\infty }f(n)=\int _{0}^{\infty }\left(f(t)+{\frac {i\,f(it)-i\,f(-it)}{e^{2\pi t}-1}}\right)\,dt.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fc63c08214cbfdffe91fc160d22ef651168609b8)
The case ƒ(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
References
- ^ Hermite, C. (1901). "Extrait de quelques lettres de M. Ch. Hermite à M. S. Píncherle". Annali di Matematica Pura ed Applicata, Serie III. 5: 57–72.
- ^ "Summation Formulas of Euler-Maclaurin and Abel-Plana: Old and New Results and Applications" (PDF).
- 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
External links
- Anderson, David. "Abel-Plana Formula". MathWorld.