# 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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