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.

Formal statement

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Suppose that   is an analytic function in a region in the complex plane   which contains the closed disk   of radius   about the origin,   are the zeros of   in the interior of   (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   in the interior of   and the average of   on the boundary circle  , and can be seen as a generalisation of the mean value property of harmonic functions. Namely, if   has no zeros in  , 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.

Proof[1]

It suffices to prove the case for  .

  1. If   contains zeros on the circle boundary, then we can define  , where   are the zeros on the circle boundary. Since   we have reduced to proving the theorem for  , that is, the case with no zeros on the circle boundary.
  2. Define   and fill in all the removable singularities. We obtain a function   that is analytic in  , and it has no roots in  .
  3. Since   is a harmonic function, we can apply Poisson integral formula to it, and obtain   where   can be written as  
  4. Now,   is a multiple of a contour integral of function   along a circle of radius  . Since   has no poles in  , the contour integral is zero.

Applications

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Jensen's formula can be used to estimate the number of zeros of an analytic function in a circle. Namely, if   is a function analytic in a disk of radius   centered at   and if   is bounded by   on the boundary of that disk, then the number of zeros of   in a circle of radius   centered at the same point   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]

Generalizations

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Jensen's formula may be generalized for functions which are merely meromorphic on  . Namely, assume that

 

where   and   are analytic functions in   having zeros at   and   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. If   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. If the function   has no zeros in the unit disk, the Poisson-Jensen formula reduces to

 

which is the Poisson formula for the harmonic function  .

See also

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References

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  1. ^ a b Ahlfors, Lars V. (1979). "5.3.1, Jensen's formula". Complex analysis : an introduction to the theory of analytic functions of one complex variable (3rd ed.). New York: McGraw-Hill. ISBN 0-07-000657-1. OCLC 4036464.
  2. ^ Paley & Wiener 1934, pp. 14–20.
  3. ^ Sayed & Kailath 2001, pp. 469–470.

Sources

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