Abel–Plana formula

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

[math]\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 }[/math]

For the case [math]\displaystyle{ a=0 }[/math] we have

[math]\displaystyle{ \sum_{n=0}^\infty f(n)=\frac 1 2 f(0)+ \int_0^\infty f(x) \, dx+ i \int_0^\infty \frac{f(i t)-f(-i t)}{e^{2\pi t}-1} \, dt. }[/math]

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).

An example is provided by the Hurwitz zeta function,

[math]\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}, }[/math]

which holds for all [math]\displaystyle{ s \in \mathbb{C} }[/math], s ≠ 1. Another powerful example is applying the formula to the function [math]\displaystyle{ e^{-n}n^{x} }[/math]: we obtain

[math]\displaystyle{ \Gamma(x+1)=\operatorname{Li}_{-x}\left(e^{-1}\right)+\theta(x) }[/math] where [math]\displaystyle{ \Gamma(x) }[/math] is the gamma function, [math]\displaystyle{ \operatorname{Li}_{s}\left(z\right) }[/math] is the polylogarithm and [math]\displaystyle{ \theta(x)=\int_{0}^{\infty}\frac{2t^{x}}{e^{2\pi t}-1}\sin\left(\frac{\pi x}{2}-t\right)dt }[/math].

Abel also gave the following variation for alternating sums:

[math]\displaystyle{ \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, }[/math]

which is related to the Lindelöf summation formula

[math]\displaystyle{ \sum_{k=m}^\infty (-1)^kf(k)=(-1)^m\int_{-\infty}^\infty f(m-1/2+ix)\frac{dx}{2\cosh(\pi x)}. }[/math]

Proof

Let [math]\displaystyle{ f }[/math] be holomorphic on [math]\displaystyle{ \Re(z) \ge 0 }[/math], such that [math]\displaystyle{ f(0) = 0 }[/math], [math]\displaystyle{ f(z) = O(|z|^k) }[/math] and for [math]\displaystyle{ \operatorname{arg}(z)\in (-\beta,\beta) }[/math], [math]\displaystyle{ f(z) = O(|z|^{-1-\delta}) }[/math]. Taking [math]\displaystyle{ a=e^{i \beta/2} }[/math] with the residue theorem [math]\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). }[/math]

Then [math]\displaystyle{ \begin{align} \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{align} }[/math]

Using the Cauchy integral theorem for the last one. [math]\displaystyle{ \int_0^{a\infty} \frac{f(z)}{e^{-2i\pi z}-1} \, dz = \int_0^\infty \frac{f(at)}{e^{-2i\pi a t}-1} \, d(at), }[/math] thus obtaining [math]\displaystyle{ \sum_{n=0}^\infty f(n)=\int_0^\infty \left(f(t)+\frac{a\, f(a t)}{e^{-2i\pi a t}-1} + \frac{a^{-1} f(a^{-1}t)}{e^{2i\pi a^{-1} t}-1}\right) \, dt. }[/math]

This identity stays true by analytic continuation everywhere the integral converges, letting [math]\displaystyle{ a\to i }[/math] we obtain the Abel–Plana formula [math]\displaystyle{ \sum_{n=0}^\infty f(n)=\int_0^\infty \left(f(t)+\frac{i\, f(i t)-i\, f(-it)}{e^{2\pi t}-1}\right) \, dt. }[/math]

The case ƒ(0) ≠ 0 is obtained similarly, replacing [math]\displaystyle{ \int_{a^{-1}\infty}^{a\infty} \frac{f(z)}{e^{-2i\pi z}-1} \, dz }[/math] by two integrals following the same curves with a small indentation on the left and right of 0.

See also

• Euler–Maclaurin summation formula
• Euler–Boole summation
• Ramanujan summation

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 
• Olver, Frank William John (1997), Asymptotics and special functions, AKP Classics, Wellesley, MA: A K Peters Ltd., ISBN 978-1-56881-069-0 
• 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". http://mathworld.wolfram.com/Abel-PlanaFormula.html.