In complex analysis, Jensen's formula relates the average magnitude of an analytic function on a circle with the number of its zeros inside the circle.
The formula was introduced by Johan Jensen (1899) and forms an important statement in the study of entire functions.
is an analytic function in a region in the complex plane
(repeated according to their respective multiplicity), and that
Jensen's formula states that[1] This formula establishes a connection between the moduli of the zeros of
on the boundary circle
, and can be seen as a generalisation of the mean value property of harmonic functions.
, then Jensen's formula reduces to which is the mean-value property of the harmonic function
An equivalent statement of Jensen's formula that is frequently used is where
denotes the number of zeros of
in the disc of radius
centered at the origin.
It suffices to prove the case for
Jensen's formula can be used to estimate the number of zeros of an analytic function in a circle.
is a function analytic in a disk of radius
on the boundary of that disk, then the number of zeros of
in a circle of radius
does not exceed Jensen's formula is an important statement in the study of value distribution of entire and meromorphic functions.
In particular, it is the starting point of Nevanlinna theory, and it often appears in proofs of Hadamard factorization theorem, which requires an estimate on the number of zeros of an entire function.
Jensen's formula is also used to prove a generalization of Paley-Wiener theorem for quasi-analytic functions with
[2] In the field of control theory (in particular: spectral factorization methods) this generalization is often referred to as the Paley–Wiener condition.
[3] Jensen's formula may be generalized for functions which are merely meromorphic on
Namely, assume that where
are analytic functions in
respectively, then Jensen's formula for meromorphic functions states that Jensen's formula is a consequence of the more general Poisson–Jensen formula, which in turn follows from Jensen's formula by applying a Möbius transformation to
It was introduced and named by Rolf Nevanlinna.
is a function which is analytic in the unit disk, with zeros
located in the interior of the unit disk, then for every
in the unit disk the Poisson–Jensen formula states that Here, is the Poisson kernel on the unit disk.
has no zeros in the unit disk, the Poisson-Jensen formula reduces to which is the Poisson formula for the harmonic function