Jensen's formula

Jensen's formula is a special rule used in complex analysis. It helps us understand how numbers change when we use certain math rules on circles. This idea connects the size of a math rule on the edge of a circle to how many times the rule hits zero inside that circle.

The formula was created by a mathematician named Johan Jensen. It is very useful when studying a type of math rule that never stops or ends, called entire functions. This formula helps mathematicians learn more about these special rules and how they behave.

Formal statement

Jensen's formula is a special way to study functions in complex analysis. It helps us learn how the values of a function change on a circle and how many times the function crosses zero inside that circle.

The formula connects two things: the number of zeros a function has inside a circle, and the average value of the function’s size on the edge of the circle. This makes it useful for learning about functions that are smooth and follow certain rules.

Applications

Jensen's formula helps us learn about how many times a special math rule, called an analytic function, equals zero inside a circle. This is useful when studying functions that are defined everywhere.

The formula is an important idea in a branch of math called Nevanlinna theory. It also helps prove key theorems, like the Hadamard factorization theorem and a version of the Paley-Wiener theorem for some functions. In control theory, this idea is known as the Paley–Wiener condition.

Main articles: Nevanlinna theory, Hadamard factorization theorem, Paley-Wiener theorem, quasi-analytic functions, control theory, spectral factorization methods

Generalizations

Jensen's formula can be used for special types of functions called meromorphic functions. These functions can be written in a special way, and the formula helps us learn more about them.

The formula is related to another idea called the Poisson–Jensen formula. This formula was introduced by Rolf Nevanlinna. It helps explain how some functions act inside a circle, using something called the Poisson kernel.