In mathematics, the Abel–Plana formula is a summation formula discovered independently by Niels Henrik Abel (1823) and Giovanni Antonio Amedeo Plana (1820).
2 π x
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.
An example is provided by the Hurwitz zeta function, which holds for all
Another powerful example is applying the formula to the function
is the gamma function,
2 π t
π x
Abel also gave the following variation for alternating sums: which is related to the Lindelöf summation formula [2] Let
arg ( z ) ∈ ( − β , β )
with the residue theorem
− 2 i π z
d z = − 2 i π
− 2 i π z
− 2 i π z
− 2 i π z
2 i π z
2 i π
{\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}}}
Using the Cauchy integral theorem for the last one.
− 2 i π z
− 2 i π a t
− 2 i π a t
2 i π
This identity stays true by analytic continuation everywhere the integral converges, letting
we obtain the Abel–Plana formula
2 π t
The case ƒ(0) ≠ 0 is obtained similarly, replacing
− 2 i π z
by two integrals following the same curves with a small indentation on the left and right of 0.