Nevanlinna theory

The Nevanlinna theory, named after its founder Rolf Nevanlinna, belongs to the mathematical branch of function theory. She tells about the distribution of values ​​of meromorphic functions.

Overview

Basic idea of ​​the Nevanlinna theory (or value distribution theory ) is to gain a quantitative version of the theorem of Picard. This theorem states that there are no non-constant meromorphic function for different values ​​of the Riemann sphere. To gain a quantitative version of this theorem, we look for and the number of sites of a non- constant meromorphic function in the closed circle around 0 with radius. The sites are counted according to multiplicity. It turns out to be appropriate, instead of the function the integrated number function

Look at. ( For this must be modified slightly, see below. ) Nevanlinna now defined a characteristic function, which tends to infinity, and showed that for most values ​​of the functions and of the same order. More imply its two main sets that

For all and

For different, in comparison with a very small error term. The Picardsche sentence follows from this immediately.

The Nevanlinna characteristic

Thus, the function defining the integral exists also for practice, we define the number function more precise than that given above by

Obviously applies to all generations. Short to write well, making for. Furthermore, we define the Schmiegungsfunktion by

Said. For one set accordingly. The Nevanlinna characteristic is then defined by

It applies when is not constant. Is transcendent, even applies

For entire functions is the maximum amount

A measure for the growth of the function. For valid

The order of a meromorphic function is defined by

For entire functions can be replaced by because of the above relationship between Nevanlinna characteristic and maximum amount here. Functions of finite order form an important and extensively studied class of meromorphic functions.

Alternatively to the Nevanlinna characteristic can also use an introduced by Lars Valerian Ahlfors and Shimizu Tatsujirō variant. The Ahlfors - Shimizu characteristic differs only to a limited Term of the Nevanlinna characteristic

The Nevanlinnaschen main theorems

The First Law states that for all

Applies. In particular we

The first main theorem is a simple consequence of Jensen's formula.

Much deeper is the Second Law. This means that for different inequality

Holds, where

And is smaller compared to the error term. More precisely, that a lot of finite measurement exists so that

For, .

Using the first law of thermodynamics can be seen that the inequality

An equivalent formulation of the second law is.

The term is one of the multiple points of the function. Denoting with and and corresponding functions, but also multiple - points are counted only once, we obtain

The defect relation

One of the main consequences of the second law is the deficiency relation. For called

Nevanlinnadefekt of. The second equal sign is paid by the first law, since for. (It is always that is not constant. ) Follows from the first law that for all. This is called defective value or Nevanlinnaschen exception value if true. According to the second law, the amount of the defective value is countable and it is the defect relation

Wherein the sum of all defect values ​​is formed. The defect relation is a far-reaching generalization of the theorem of Picard, as is transcendent and takes the value of only finitely often, so true. Also, a given Borel tightening the set of Picard follows easily from the second law.

More results in defects

A central problem of Nevanlinnatheorie was a long time whether the defect relation and the inequality are the only restrictions on the Nevanlinnadefekte of a meromorphic function. This so-called inverse problem of Nevanlinnatheorie was solved in 1976 by David Drasin. For functions of finite order there are various other restrictions. For example equality in the defect relation, it follows by a natural number. This had been suggested by Rolf Nevanlinna brother Frithiof and was proved in 1987 by Drasin. As a further result about meromorphic functions of finite order Nevanlinnadefekte is an example of a result of Allen Weitsman called, who showed in 1972 that for such functions

Applies.

Many more results Nevanlinnadefekten to be found in the books listed below, with the book by Goldberg and Ostrovskii includes an appendix by A. Eremenko and JK Langley, in which recent developments are presented.

Applications

The Nevanlinnatheorie has found applications in various fields. So it has proved essential tool in the study of differential equations and functional equations in the complex domain, see for example the books by Jank - Volkmann and Laine.

Nevanlinna proved as one of the first applications of his theory following uniqueness theorem: If the two - points of meromorphic functions and for 5 values ​​match, the following applies. This sentence was the starting point for many other sets of this type.

More recently met analogies found by Paul Vojta between Nevanlinnatheorie and Diophantine approximation with great interest, see the book by Ru.

Generalizations

This article is limited to the classical theory of a complex variable. There are several generalizations about on algebroide functions holomorphic curves, functions of several complex variables and quasiregular mappings.