# Value distribution theory of holomorphic functions

In

mathematics, the

**value distribution theory of holomorphic functions** is a division of

mathematical analysis. It tries to get quantitative measures of the number of times a function

*f*(

*z*) assumes a value

*a*, as

*z* grows in size, refining the Picard theorem on behaviour close to an

essential singularity. The theory exists for analytic functions (and meromorphic functions) of one complex variable

*z*, or of

several complex variables.

In the case of one variable the term **Nevanlinna theory** is also common. The now-classical theory received renewed interest, when Paul Vojta suggested some analogies with the problem of integral solutions to Diophantine equations. These turned out to involve some close parallels, and to lead to fresh points of view on the Mordell conjecture and related questions.