Jensen's formula

Jensen's formula is a special rule used in a part of math called complex analysis. It helps us understand how big numbers get 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 understand how the values of a function behave 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 understand how many times a special kind of math rule, called an analytic function, equals zero inside a circle. This is useful in studying functions that are defined everywhere, known as entire functions.

The formula is a key idea in a branch of math called Nevanlinna theory. It also helps prove important theorems, like the Hadamard factorization theorem and a version of the Paley-Wiener theorem for certain types of functions. In control theory, this related 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 also apply to special types of functions called meromorphic functions. These functions can be written in a particular form, and the formula helps us understand their behavior.

The formula is connected to another important idea called the Poisson–Jensen formula. This formula builds on Jensen's work and was later introduced by Rolf Nevanlinna. It helps describe how certain functions behave inside a circle, using a tool called the Poisson kernel.