Jensen's formula
Encyclopedia
In the mathematical field known as complex analysis
, Jensen's formula, named after Johan Jensen, relates the average magnitude of an analytic function
on a circle with the magnitudes of its zeros inside the circle. It forms an important statement in the study of entire function
s.
which contains the closed disk D of radius r about the origin, a1, a2, ..., an 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 function ƒ inside the disk |z| < r and the average of its modulus |ƒ(z)| on the boundary circle |z| = r, and can be seen as a generalisation of the mean value property of harmonic function
s. In its turn, Jensen's formula may be generalised to the Poisson–Jensen formula, which provides a similar connection for functions that are merely meromorphic in a region containing the disk.
Complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including number theory and applied mathematics; as well as in physics,...
, Jensen's formula, named after Johan Jensen, relates the average magnitude of an analytic function
Analytic function
In mathematics, an analytic function is a function that is locally given by a convergent power series. There exist both real analytic functions and complex analytic functions, categories that are similar in some ways, but different in others...
on a circle with the magnitudes of its zeros inside the circle. It forms an important statement in the study of entire function
Entire function
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic over the whole complex plane...
s.
The statement
Suppose that ƒ is an analytic function in a region in the complex planeComplex plane
In mathematics, the complex plane or z-plane is a geometric representation of the complex numbers established by the real axis and the orthogonal imaginary axis...
which contains the closed disk D of radius r about the origin, a1, a2, ..., an 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 function ƒ inside the disk |z| < r and the average of its modulus |ƒ(z)| on the boundary circle |z| = r, and can be seen as a generalisation of the mean value property of harmonic function
Harmonic function
In mathematics, mathematical physics and the theory of stochastic processes, a harmonic function is a twice continuously differentiable function f : U → R which satisfies Laplace's equation, i.e....
s. In its turn, Jensen's formula may be generalised to the Poisson–Jensen formula, which provides a similar connection for functions that are merely meromorphic in a region containing the disk.