---
tag: 1EHH63VNAD
kind: concept
name: Nevanlinna theory
msc: ["30D35"]
sources:
  - url: https://en.wikipedia.org/wiki/Nevanlinna_theory
same_as:
  wikidata: Q1281926
  wikipedia: Nevanlinna theory
---

## Summary

Nevanlinna theory is the quantitative study of how often a [[meromorphic function|0NSD1TPHR0]] 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|YT6Z0GWS0W]].
The theory refines [[Picard's little theorem|R2PHM118NG]] and has analogues in Diophantine approximation.

## Explanations

### L1 human

A nonconstant polynomial of degree $d$ takes every value exactly $d$ times.
An entire or meromorphic function can take a value infinitely often, or never, as $e^z$ never takes $0$.
Nevanlinna theory replaces the degree by a function $T(r, f)$ that grows with the radius $r$.
The [[first main theorem|QJWCFAK4AK]] says that each value is taken at most about $T(r, f)$ times, counted with a logarithmic weight.
The [[second main theorem|GWYE6745JK]] says that most values are taken about that often, and that at most two values can be strongly avoided.