Jensen's formula

Jensen's formula

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) e 0, then ::log |f(0)| = -sum_{k=1}^n logleft(frac{r} ight)+frac{1}{2pi}int_0^{2pi}log|f(re^{i heta})|d heta.This formula establishes a connection between the moduli of the zeros of the function "f" inside the disk |z| 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

*cite book | author = L. V. Ahlfors| title = Complex Analysis | publisher = McGraw-Hill | year = 1979 | id=ISBN 0-07-000657-1


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