Jensen's formula

From Wikipedia, the free encyclopedia

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

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.

[edit] References


This mathematical analysis-related article is a stub. You can help Wikipedia by expanding it.
Languages