- Jensen's formula
Jensen's formula (after
Johan Jensen ) incomplex analysis relates the behaviour of ananalytic function on a circle with the moduli of the zeros inside the circle, and is important in the study ofentire function s.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 function s. Jensen's formula in turn may be generalised to give the Poisson-Jensen formula, which gives a similar result for functions which are merelymeromorphic 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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