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. Its homeomorphic solutions are exactly the quasiconformal mappings with complex dilatation $\mu$. The measurable Riemann mapping theorem says that it always has such a solution.