Jensen's formula

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Jensen's formula (after Johan Jensen) in complex analysis relates the behaviour of an analytic function on a circle with the moduli of the zeros inside the circle, and is important in the study of entire functions.

The statement of Jensen's formula is

f

is an analytic function in a region which contains the closed disk D in the complex plane, if $a_1, a_2,\dots,a_n$ are the zeros of

f

in the interior of D repeated according to multiplicity, and if $f(0)\ne 0$ , then

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

This formula establishes a connection between the moduli of the zeros of the function f inside the disk $| z | < r$ and the values of $| f (z) |$ on the circle $| z | = r$ , and can be seen as a generalisation of the mean value property of harmonic functions. Jensen's formula in turn may be generalised to give the Poisson-Jensen formula, which gives a similar result for functions which are merely meromorphic in a region containing the disk.