User:Madmath789/jensen

Jensen's formula 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


 * If $$f$$ 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.