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 $f(0)\neq 0$, 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.

References

• L. V. Ahlfors (1979). Complex Analysis. McGraw-Hill. ISBN 0-070-00657-1.

{{maths-stub}} [[Category:Complex analysis]]