Abel–Plana formula

Not to be confused with Abel's summation 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

$\sum _{n=0}^{\infty }f\left(a+n\right)=\int _{a}^{\infty }f\left(x\right)dx+{\frac {f\left(a\right)}{2}}+\int _{0}^{\infty }{\frac {f\left(a-ix\right)-f\left(a+ix\right)}{i\left(e^{2\pi x}-1\right)}}dx$

For the case $a=0$ we have

$\sum _{n=0}^{\infty }f(n)={\frac {1}{2}}f(0)+\int _{0}^{\infty }f(x)\,dx+i\int _{0}^{\infty }{\frac {f(it)-f(-it)}{e^{2\pi t}-1}}\,dt.$

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,

$\zeta (s,\alpha )=\sum _{n=0}^{\infty }{\frac {1}{(n+\alpha )^{s}}}={\frac {\alpha ^{1-s}}{s-1}}+{\frac {1}{2\alpha ^{s}}}+2\int _{0}^{\infty }{\frac {\sin \left(s\arctan {\frac {t}{\alpha }}\right)}{(\alpha ^{2}+t^{2})^{\frac {s}{2}}}}{\frac {dt}{e^{2\pi t}-1}},$

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

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

Abel also gave the following variation for alternating sums:

$\sum _{n=0}^{\infty }(-1)^{n}f(n)={\frac {1}{2}}f(0)+i\int _{0}^{\infty }{\frac {f(it)-f(-it)}{2\sinh(\pi t)}}\,dt,$

which is related to the Lindelöf summation formula

$\sum _{k=m}^{\infty }(-1)^{k}f(k)=(-1)^{m}\int _{-\infty }^{\infty }f(m-1/2+ix){\frac {dx}{2\cosh(\pi x)}}.$

Proof

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

$$\int _{a^{-1}\infty }^{0}+\int _{0}^{a\infty }{\frac {f(z)}{e^{-2i\pi z}-1}}\,dz=-2i\pi \sum _{n=0}^{\infty }\operatorname {Res} \left({\frac {f(z)}{e^{-2i\pi z}-1}}\right)=\sum _{n=0}^{\infty }f(n).$$

Then

$${\begin{aligned}\int _{a^{-1}\infty }^{0}{\frac {f(z)}{e^{-2i\pi z}-1}}\,dz&=-\int _{0}^{a^{-1}\infty }{\frac {f(z)}{e^{-2i\pi z}-1}}\,dz\\&=\int _{0}^{a^{-1}\infty }{\frac {f(z)}{e^{2i\pi z}-1}}\,dz+\int _{0}^{a^{-1}\infty }f(z)\,dz\\&=\int _{0}^{\infty }{\frac {f(a^{-1}t)}{e^{2i\pi a^{-1}t}-1}}\,d(a^{-1}t)+\int _{0}^{\infty }f(t)\,dt.\end{aligned}}$$

Using the Cauchy integral theorem for the last one.

$$\int _{0}^{a\infty }{\frac {f(z)}{e^{-2i\pi z}-1}}\,dz=\int _{0}^{\infty }{\frac {f(at)}{e^{-2i\pi at}-1}}\,d(at),$$

thus obtaining

$$\sum _{n=0}^{\infty }f(n)=\int _{0}^{\infty }\left(f(t)+{\frac {a\,f(at)}{e^{-2i\pi at}-1}}+{\frac {a^{-1}f(a^{-1}t)}{e^{2i\pi a^{-1}t}-1}}\right)\,dt.$$

This identity stays true by analytic continuation everywhere the integral converges, letting $a\to i$ we obtain the Abel–Plana formula

$$\sum _{n=0}^{\infty }f(n)=\int _{0}^{\infty }\left(f(t)+{\frac {i\,f(it)-i\,f(-it)}{e^{2\pi t}-1}}\right)\,dt.$$

The case ƒ(0) ≠ 0 is obtained similarly, replacing ${\textstyle \int _{a^{-1}\infty }^{a\infty }{\frac {f(z)}{e^{-2i\pi z}-1}}\,dz}$ 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, 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.