Nevanlinna theory

In the mathematical field of complex analysis, Nevanlinna theory is part of the theory of meromorphic functions. It was devised in 1925, by Rolf Nevanlinna. Hermann Weyl has called it "one of the few great mathematical events of (the twentieth) century." The theory describes the asymptotic distribution of solutions of the equation ƒ(z) = a, as a varies. A fundamental tool is the Nevanlinna characteristic T(r, ƒ) which measures the rate of growth of a meromorphic function.

Other main contributors in the first half of the 20th century were Lars Ahlfors, André Bloch, Henri Cartan, Edward Collingwood, Otto Frostman, Frithiof Nevanlinna, Henrik Selberg, Tatsujiro Shimizu, Oswald Teichmüller, and Georges Valiron. In its original form, Nevanlinna theory deals with meromorphic functions of one complex variable defined in a disc |z| ≤ R or in the whole complex plane (R = ∞). Subsequent generalizations extended Nevanlinna theory to algebroid functions, holomorphic curves, holomorphic maps between complex manifolds of arbitrary dimension, quasiregular maps and minimal surfaces.

This article describes mainly the classical version for meromorphic functions of one variable, with emphasis on functions meromorphic in the complex plane. General references for this theory are Goldberg & Ostrovskii, Hayman and Lang (1987).

Let f be a meromorphic function. For every r ≥ 0, let n(r,f) be the number of poles, counting multiplicity, of the meromorphic function f in the disc |z| ≤ r. Then define the Nevanlinna counting function by

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