# 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) $d_{f}=\sup_{z}\lvert\mu(z)\rvert$.

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