Nevanlinna theory

Summary

Nevanlinna theory is the quantitative study of how often a meromorphic function on $\C$ takes each value. For each value $a$ it uses a counting function, which counts the solutions of $f(z) = a$ in a disc. It also uses a proximity function, which measures how close $f$ comes to $a$ on the boundary circle. Its two main theorems compare these quantities with a single growth measure, the Nevanlinna characteristic. The theory refines Picard's little theorem and has analogues in Diophantine approximation.