In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, the value distribution theory of holomorphic functions is a division of mathematical analysis

Mathematical analysis

Mathematical analysis, which mathematicians refer to simply as analysis, has its beginnings in the rigorous formulation of infinitesimal calculus. It is a branch of pure mathematics that includes the theories of differentiation, integration and measure, limits, infinite series, and analytic functions...

. 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

Picard theorem

In complex analysis, the term Picard theorem refers to either of two distinct yet related theorems, both of which pertain to the range of an analytic function.-Little Picard:...

 on behaviour close to an essential singularity

Essential singularity

In complex analysis, an essential singularity of a function is a "severe" singularity near which the function exhibits extreme behavior.The category essential singularity is a "left-over" or default group of singularities that are especially unmanageable: by definition they fit into neither of the...

. The theory exists for 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...

s (and meromorphic function

Meromorphic function

In complex analysis, a meromorphic function on an open subset D of the complex plane is a function that is holomorphic on all D except a set of isolated points, which are poles for the function...

s) of one complex variable z, or of several complex variables

Several complex variables

The theory of functions of several complex variables is the branch of mathematics dealing with functionson the space Cn of n-tuples of complex numbers...

.

In the case of one variable the term Nevanlinna theory

Nevanlinna theory

Nevanlinna theory is a branch of complex analysis developed by Rolf Nevanlinna. It deals with the value distribution theory of holomorphic functions in one variable, usually denoted z....

, after Rolf Nevanlinna

Rolf Nevanlinna

Rolf Herman Nevanlinna was one of the most famous Finnish mathematicians. He was particularly appreciated for his work in complex analysis.- The Nevanlinna family :...

, is also common. The now-classical theory received renewed interest, when Paul Vojta

Paul Vojta

Paul Alan Vojta is an American mathematician, known for his work in number theory on diophantine geometry and diophantine approximation....

 suggested some analogies with the problem of integral solutions to Diophantine equation

Diophantine equation

In mathematics, a Diophantine equation is an indeterminate polynomial equation that allows the variables to be integers only. Diophantine problems have fewer equations than unknown variables and involve finding integers that work correctly for all equations...

s. These turned out to involve some close parallels, and to lead to fresh points of view on the Mordell conjecture and related questions.