---
tag: BWZ1H4FAQG
kind: concept
name: Wiman–Valiron theory
msc: ["30D20", "30D15"]
sources:
  - url: https://en.wikipedia.org/wiki/Wiman%E2%80%93Valiron_theory
same_as:
  wikidata: Q55648775
  wikipedia: Wiman–Valiron theory
---

## Summary

Wiman–Valiron theory studies a transcendental [[entire function|0419794R0T]] through the largest term of its [[Taylor series|0225QPWRRE]] on each circle $\abs{z} = r$.
The index of this maximal term, the central index $\nu(r)$, grows to infinity with $r$.
Near points where $\abs{f}$ attains its maximum on the circle, $f$ behaves like the single monomial $a_{\nu(r)} z^{\nu(r)}$, outside a small exceptional set of radii.
A main application is to the growth of entire solutions of linear differential equations with polynomial coefficients.

## Explanations

### L1 human

An entire function is an infinite sum of terms $a_n z^n$.
On a large circle, one term is usually much bigger than the others, and its index is the central index.
Near the point where the function is largest on the circle, the function looks like this one term.
In particular, each derivative multiplies the function by about $\nu(r)/z$, as for a monomial.
For $e^z$ the central index on the circle of radius $r$ is about $r$, and indeed $(e^z)' = e^z \approx (r/z) e^z$ near $z = r$.