beurlings function

Input interpretation

Beurling's function

Definition

The entire function B(z) | = | [(sin(π z))/π]^2[2/z + sum_(n = 0)^∞ 1/(z - n)^2 - sum_(n = 1)^∞ 1/(z + n)^2] | = | 1 - (2sin^2(π z))/(π^2 z^2)[z^2 ψ_1(z) - z - 1], where ψ_1(z) is a polygamma function. It satisfies B(z) = O(e^(2π(ℑ[z]))) and B(x)>=sgn(x) for all real x. Amazingly, it also has the integral integral_(-∞)^∞[B(x) - sgn(x)] d x = 1. Furthermore, among all functions with the first two properties, B(x) minimizes the integral (-1).
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Subject classifications

MathWorld

polygamma functions

MSC 2010

33B15
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