Jensen's formula

In the mathematical field known as complex analysis, Jensen's formula, introduced by Johan Jensen (1899), relates the average magnitude of an analytic function on a circle with the number of its zeros inside the circle. It forms an important statement in the study of entire functions.

Suppose that 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. Jensen's formula states that

This formula establishes a connection between the moduli of the zeros of the function ƒ inside the disk D and the average of log |f(z)| on the boundary circle |z| = r, and can be seen as a generalisation of the mean value property of harmonic functions. Namely, if f has no zeros in D, then Jensen's formula reduces to

which is the mean-value property of the harmonic function ${\displaystyle \log |f(z)|}$.

An equivalent statement of Jensen's formula that is frequently used is

where ${\displaystyle n(t)}$ denotes the number of zeros of ${\displaystyle f}$ in the disc of radius ${\displaystyle t}$ centered at the origin.

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