Jensen's formula: Difference between revisions

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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

Suppose ƒ is an analytic function in a region in the complex plane which contains the closed disk D of radius r about the origin, a₁, a₂, ..., a_n are the zeros of ƒ in the interior of D repeated according to multiplicity, and ƒ(0) ≠ 0. Then

${\displaystyle \log |f(0)|=-\sum _{k=1}^{n}\log \left({\frac {r}{|a_{k}|}}\right)+{\frac {1}{2\pi }}\int _{0}^{2\pi }\log |f(re^{i\theta })|\,d\theta .}$

This formula establishes a connection between the moduli of the zeros of the function ƒ inside the disk |z| < r and the values of |ƒ(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.

• L. V. Ahlfors (1979). Complex Analysis. McGraw–Hill. ISBN 0-07-000657-1.