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.

Nevanlinna theory is very useful when dealing with meromorphic functions as their poles can block attempts at analysis by more conventional means such as the maximum modulus principle.

Basic definitions

We first define n(r, ƒ) to be the number of poles of ƒ in the disc |z| < r, with regards to multiplicity — i.e. a double pole will add 2 to n(r, ƒ) and a simple pole will add 1. Hence we set:

N(r,f) = \int\limits_0^r\left( n(t,f) - n(0,f) \right)\dfrac{dt}{t} + n(0,f)\log r.

This function, known as the N function, counts the poles of f. We further define \scriptstyle\overline{N}(r,\,f) as counting the poles without regards to multiplicity.

We now define the second Nevanlinna functional:

m(r,f) = \dfrac{1}{2\pi}\int\limits_0^{2\pi}\log^+\left|f(re^{i\theta})\right|\,d\theta,

and hence the Nevanlinna characteristic function:

T(r,f) = m(r,f) + N(r,f) = \dfrac{1}{2\pi}\int\limits_0^{2\pi}\log^+\left|f(re^{i\theta})\right|d\theta + \int\limits_0^r\left( n(t,f) - n(0,f) \right)\dfrac{dt}{t} + n(0,f)\log r.

References

  • S. Lang (1997). Survey of Diophantine geometry. Springer-Verlag. pp. 192–204. ISBN 3-540-61223-8. 
  • Min Ru (2001). Nevanlinna Theory and Its Relation to Diophantine Approximation. World Scientific Publishing. ISBN 9810244029.