Abel–Plana 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 ∑ n = 0 ∞ f ( a + n ) = f ( a ) 2 + ∫ a ∞ f ( x ) d x + i ∫ 0 ∞ f ( a + i t ) − f ( a − i t ) e 2 π t − 1 d t {\displaystyle \sum _{n=0}^{\infty }f\left(a+n\right)={\frac {f\left(a\right)}{2}}+\int _{a}^{\infty }f\left(x\right)dx+i\int _{0}^{\infty }{\frac {f\left(a+it\right)-f\left(a-it\right)}{e^{2\pi t}-1}}dt} For the case a = 0 {\displaystyle a=0} we have ∑ n = 0 ∞ f ( n ) = f ( 0 ) 2 + ∫ 0 ∞ f ( x ) d x + i ∫ 0 ∞ f ( i t ) − f ( − i t ) e 2 π t − 1 d t .