Abel-Plana formula

Discovered independently by Niels Henrik Abel (1823) and Giovanni Antonio Amedeo Plana (1820).

the Abel-Plana formula

1. (mathematics) A summation formula stating that ${\displaystyle \sum _{n=0}^{\infty }f\left(a+n\right)=\int _{a}^{\infty }f\left(x\right)dx+{\frac {f\left(a\right)}{2}}+\int _{0}^{\infty }{\frac {f\left(a-ix\right)-f\left(a+ix\right)}{i\left(e^{2\pi x}-1\right)}}dx}$.