Abel-Plana Formula

Theorem

Let $\map f z$ be analytic for real part of $\map \Re z \ge 0$.

Suppose that either $\ds \sum_{n \mathop = 0}^\infty \map f n$ converges or $\ds \int_0^\infty \map f x \rd x$ converges.

Assume further that:

$\ds \lim_{y \mathop \to \infty} \size {\map f {x \pm i y} } e^{-2 \pi y} = 0$ uniformly in $x$ on every finite interval

$\ds \int_0^{\infty} \size {\map f {x \pm i y} } e^{-2 \pi y} \rd y$ exists for every $x \ge 0$ and tends to $0$ as $x \to \infty$.

Then:

$\ds \sum_{n \mathop = 0}^\infty \map f n = \int_0^\infty \map f x \rd x + \dfrac 1 2 \map f 0 + i \int_0^\infty \dfrac {\map f {i t} - \map f {-i t} } {e^{2 \pi t} - 1} \rd t$

Proof

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Source of Name

This entry was named for Niels Henrik Abel and Giovanni Antonio Amedeo Plana.

Sources

• 1920: E.T. Whittaker and G.N. Watson: A Course of Modern Analysis (3rd ed.): $7$: The Process of Analysis
• Weisstein, Eric W. "Abel-Plana Formula." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Abel-PlanaFormula.html