Abel Plana empirical formula

The Abel – Plana sum formula is a sum formula that was independently discovered by Niels Henrik Abel (1823) and Giovanni Antonio Amedeo Plana (1820). It says that

${\displaystyle \sum \limits _{n=0}^{\infty }f(n)=\int \limits _{0}^{\infty }f(t)\,\mathrm {d} t+{\frac {1}{2}}\,f(0)+i\int \limits _{0}^{\infty }{\frac {f(it)-f(-it)}{\mathrm {e} ^{2\pi t}-1}}\,\mathrm {d} t}$

is. It applies to functions f which are holomorphic in the half-plane and whose magnitude increases in a suitable manner; z. B. the assumption is sufficient ${\displaystyle \mathrm {Re} \,z>0}$

${\displaystyle |f(z)|<{\frac {C}{|z|^{1+\epsilon }}}}$

in this area for suitable constants C , ε> 0. Frank WJ Olver has even shown that the formula is valid under much weaker conditions.

The Hurwitz zeta function can be used as an example :

${\displaystyle \zeta (s,z)=\sum \limits _{n=0}^{\infty }{\frac {1}{(n+z)^{s}}}={\frac {z^{1-s}}{s-1}}+{\frac {1}{2z^{s}}}+2\int \limits _{0}^{\infty }{\frac {\sin \!\left(s\arctan {\frac {t}{z}}\right)}{(\mathrm {e} ^{2\pi t}-1){(z^{2}+t^{2})}^{\frac {s}{2}}}}\,\mathrm {d} t.}$

Abel also gave the following variant for alternating sums:

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

See also

• Euler-MacLaurine molecular formula

Individual evidence

1. ^ Abel, NH: Solution de quelquesproblemèmes à l'aide d'intégrales définies . Magazin for Naturvidenskaberne, Argang I, Bind2, Christina, 1823
2. ^ Olver, Frank WJ: Asymptotics and special functions. Reprint of the 1974 original . AKP Classics. AK Peters, Ltd., Wellesley, MA, 1997. ISBN 978-1-56881-069-0