Functions of a complex variable Definition

Beltrami equation

Builds on Wirtinger derivatives Open set Holomorphic function

Definition

Let $U$ be an open set in $\C$ . Let $\mu \colon U \to \C$ be a measurable function with $\norm{\mu}_\infty \le k$ for some constant $k < 1$ . The Beltrami equation with Beltrami coefficient $\mu$ is $f_{\bar z} = \mu f_z,$ where $f_z$ and $f_{\bar z}$ are the Wirtinger derivatives of $f$ . A solution is a function $f \in W^{1,2}_{\mathrm{loc}}(U)$ , that is, with locally square-integrable weak first partial derivatives, that satisfies the equation almost everywhere on $U$ .

Remarks

Example: for $\mu = 0$ the equation is $f_{\bar z} = 0$ , which is the Cauchy–Riemann system, and its solutions are the holomorphic functions on $U$ . Example: for a constant $c$ with $\abs{c} < 1$ , the function $f(z) = z + c \bar z$ has $f_z = 1$ and $f_{\bar z} = c$ , so it solves the equation with $\mu = c$ . Non-example: the coefficient $\mu = 1$ fails the condition $\norm{\mu}_\infty < 1$ . Then the equation reads $f_{\bar z} - f_z = i f_y = 0$ , and every function of $x$ alone solves it, so the equation is degenerate and its solutions are never homeomorphisms. A homeomorphic solution is a quasiconformal mapping whose complex dilatation is $\mu$ .

Reading path

To understand this page from the ground up, read these 4 pages in order. The foundations come first.

Show the path as a graph

Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

no evidence level

Where this leads

1 page builds on this one.

Measurable Riemann mapping theorem

Tag	`4K6B00NYCD`
URI	`m2://4K6B00NYCD`
Kind	definition
Writing	W0 stub
MSC	30C62
Defines	Beltrami equation
Uses	Wirtinger derivatives, Open set, Holomorphic function
Changed	2 November 2026

Beltrami equation

Definition

Remarks

Reading path

Where this leads

Further reading

What this page needs for W1

What links here

Used by