# Beltrami differential equation

Suppose that $\mu:G\subset{\mathbb{C}}\rightarrow{\mathbb{C}}$ is a measurable function, then the partial differential equation

 $f_{\bar{z}}(z)=\mu(z)f_{z}(z)$

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 (http://planetmath.org/QuasiconformalMapping) $\mu(z)$ and maximal small dilatation (http://planetmath.org/QuasiconformalMapping) $d_{f}=\sup_{z}\lvert\mu(z)\rvert$.

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

 $\displaystyle f_{z}$ $\displaystyle=\frac{1}{2}(u_{x}+v_{y})+\frac{i}{2}(v_{x}-u_{y}),$ $\displaystyle f_{\bar{z}}$ $\displaystyle=\frac{1}{2}(u_{x}-v_{y})+\frac{i}{2}(v_{x}+u_{y}).$

## References

• 1 L. V. Ahlfors. . Van Nostrand-Reinhold, Princeton, New Jersey, 1966
Title Beltrami differential equation BeltramiDifferentialEquation 2013-03-22 14:08:34 2013-03-22 14:08:34 jirka (4157) jirka (4157) 8 jirka (4157) Definition msc 35F20 msc 30C62 QuasiconformalMapping