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Abel–Plana formula

Niels Abel.

The Abel–Plana formula is a summation formula discovered independently by Niels Abel and Giovanni Plana. It states that

∑n=0∞f(n)=12f(0)+∫0∞f(x)dx+i∫0∞f(it)−f(−it)e2πt−1dt\displaystyle \sum_{n=0}^\infty f \left( n \right) = \frac{1}{2} f \left( 0 \right) + \int_0^\infty f \left( x \right) \mathrm{d}x + i \int_0^\infty \frac{f \left( it \right) – f \left( -it \right)}{\mathrm{e}^{2\pi t}-1} \mathrm{d}t

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