---
tag: 9J7GFJ1GPF
kind: concept
name: Beltrami equation
msc: [30C62]
sources:
  - url: https://en.wikipedia.org/wiki/Beltrami_equation
same_as:
  wikidata: Q4884794
  wikipedia: Beltrami equation
---

## Summary

The Beltrami equation is the partial differential equation $f_{\bar z} = \mu f_z$ for a function $f$ of one complex variable.
The coefficient $\mu$ is a given measurable function with $\norm{\mu}_\infty < 1$.
For $\mu = 0$ it is the Cauchy–Riemann equation, whose solutions are the [[holomorphic functions|0NAHXR1FRH]].
Its homeomorphic solutions are exactly the [[quasiconformal mappings|SB8KG46NHP]] with complex dilatation $\mu$.
The [[measurable Riemann mapping theorem|TZHD2WKNSD]] says that it always has such a solution.

## Explanations

### L1 human

The coefficient $\mu$ prescribes, at each point, an ellipse field: a direction and a stretching ratio.
A solution $f$ of the Beltrami equation is a map that sends these tiny ellipses to tiny circles.
So solving the equation means finding coordinates in which a given distorted geometry looks round.
For example, with the constant coefficient $\mu = 1/3$ the map $f(z) = z + \frac{1}{3}\bar z$ solves the equation, since $f_z = 1$ and $f_{\bar z} = 1/3$.