Nevanlinna theory

English

Etymology

Named after Finnish mathematician Rolf Nevanlinna (1895–1980), who published the theory in 1925.

Noun

Nevanlinna theory (uncountable)

1. (complex analysis) A part of the theory of meromorphic functions that describes the asymptotic distribution of solutions to the equation ƒ(z) = a, as a varies.

   A key tool in Nevanlinna theory is the Nevanlinna characteristic, $T(r,f)$, which measures the rate of growth of a meromorphic function.

   • 1992, Ilpo Laine, Nevanlinna Theory and Complex Differential Equations, Walter de Gruyter, page 1:

     Precisely, our aim has been to show how the Nevanlinna theory may be applied to get insight into the properties of solutions of complex differential equations.

   • 2001, William Cherry, Zhuan Ye, Nevanlinna's Theory of Value Distribution, Springer, page vi:

     Motivated by an analogy between Nevanlinna theory and Diophantine approximation theory, discovered independently by C. F. Osgood [Osg 1985] and P. Vojta [Vojt 1987], S. Lang recognized that the careful study of the error term in Nevanlinna'a Second Main Theorem would be of interest in itself.

   • 2010, Paul Vojta, Diophantine Approximation and Nevanlinna theory, Jean-Louis Colliot-Thélène, Peter Swinnerton-Dyer, Paul Vojta (editors), Arithmetic Geometry: Lectures given at the C.I.M.E. Summer School, Springer, Lecture Notes in Mathematics 2009, page 111,

     Beginning with the work of Osgood [65], it has been known that the branch of complex analysis known as Nevanlinna theory (also called value distribution theory) has many similarities with Roth's theorem on diophantine approximation. […] The circle of ideas has developed further in the last 20 years: Lang's conjecture on sharpening the error term in Roth's was carried over to a conjecture in Nevanlinna theory which was proved in many cases.

Synonyms

• (part of the theory of meromorphic functions): value-distribution theory

Related terms

• Nevanlinna characteristic

Translations

part of the theory of meromorphic functions

• French: théorie de Nevanlinna f
• German: Nevanlinna-Theorie f