Beltrami differential equation

Suppose that $\mu :G\subset ℂ\to ℂ$ is a measurable function, then the partial differential equation

$$f_{\overline{z}}(z)=\mu (z)f_z(z)$$

is called the Beltrami differential equation.

If furthermore andin fact $|\mu (z)|$ has a uniform bound less then 1 over the domain of definition, then the solution is a quasiconformal mapping with complex dilation (http://planetmath.org/QuasiconformalMapping) $\mu (z)$ and maximal small dilatation (http://planetmath.org/QuasiconformalMapping) $d_f={sup}_z|\mu (z)|$.

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

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

$f_z$$={\displaystyle \frac{1}{2}}(u_x+v_y)+{\displaystyle \frac{i}{2}}(v_x-u_y),$

$f_{\overline{z}}$$={\displaystyle \frac{1}{2}}(u_x-v_y)+{\displaystyle \frac{i}{2}}(v_x+u_y).$