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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 ∑ n = 0 ∞ f ( a + n ) = ∫ a ∞ f ( x ) d x + f ( a ) 2 + ∫ 0 ∞ f ( a − i x ) − f ( a + i x ) i ( e 2 π x − 1 ) d x {\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} For the case a = 0 {\displaystyle a=0} we have ∑ n = 0 ∞ f ( n ) = 1 2 f ( 0 ) + ∫ 0 ∞ f ( x ) d x + i ∫ 0 ∞ f ( i t ) − f ( − i t ) e 2 π t − 1 d t . {\displaystyle \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, ζ ( s , α ) = ∑ n = 0 https://en.wikipedia.org/wiki/Abel%E2%80%93Plana_formula

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