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( rac{r}

ight)+ rac{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.

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References

References



★ Complex Analysis, L. V. Ahlfors, , , McGraw-Hill, 1979, ISBN 0-07-000657-1

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