Jensen's formula
In the mathematical field known as complex analysis, Jensen's formula, introduced by , 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. Formal statement 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 This formula establishes a connection between the moduli of the zeros of the function inside the disk 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 Applications Jensen's formula can be used to estimate the number of zeros of an analytic function in a circle.