Jensen’s formula

Jensen’s formula relates the average magnitude of an analytic function on a circle with number of its zeroes inside the circle.

Theorem (Jensen’s formula) Suppose that f be an analytic function in a region in the complex plane which contains the closed disk D of radius r about the origin, a_1,a_2,\dots,a_n are zeros of f in the interior of D according to multiplicity, and f(0)\neq 0. Then

\displaystyle \log |f(0)|=\sum_{k=1}^n \log \frac{|a_k|}{r}+\frac{1}{2\pi}\int_0^{2\pi} \log f(re^{i\theta})\,d\theta.

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