Suppose that $\mu : G \subset {\mathbb{C}} \rightarrow {\mathbb{C}}$ is a measurable function, then the partial differential equation \begin{equation*} f_{\bar{z}}(z) = \mu(z)f_z(z) \end{equation*}is called the Beltrami differential equation.

If furthermore $\lvert \mu(z) \rvert < 1$ and in fact $\lvert \mu(z) \rvert$ has a uniform bound less then 1 over the domain of definition, then the solution is a quasiconformal mapping with complex dilation $\mu(z)$ and maximal small dilatation $d_f = \sup_z \lvert \mu(z) \rvert$ .

A conformal mapping has $f_{\bar{z}} \equiv 0$ and so the solution can be conformal if and only if $\mu \equiv 0$ .

The partial derivatives $f_z$ and $f_{\bar{z}}$ (where $\bar{z}$ is the complex conjugate of $z$ ) can here be given in terms of the real and imaginary parts of $f = u + iv$ as

Bibliography

1

L. V. Ahlfors. Lectures on Quasiconformal Mappings. Van Nostrand-Reinhold, Princeton, New Jersey, 1966