Cauchy–Riemann conditions

The Cauchy–Riemann conditions are a set of partial differential equations which, along with certain other criteria, guarantee a complex function will be holomorphic (that is, complex differentiable), since they garuntee that angles will be preserved by a mapping. Given a function $z(x+iy)=u(x,y)+i v(x,y)$ , the Cauchy–Riemann conditions are

$\frac{\part u}{\part x}=\frac{\part v}{\part y}\ ,\ \frac{\part u}{\part y}=-\frac{\part v}{\part x}$

or, using the polar representation of a complex function in terms of $r,\theta$

$\frac{\part u}{\part r}=\frac1r\frac{\part v}{\part\theta}\ ,\ \frac{\part v}{\part r}=-\frac1r\frac{\part u}{\part\theta}$

For any function which respects the Cauchy–Riemann conditions, $u, v$ will also obey Laplace's equation. This can easily be seen by differentiating a second time.

$\frac{\part^2 u}{\part x^2}+\frac{\part^2 u}{\part y^2}=\frac{\part^2 v}{\part x^2}+\frac{\part^2 v}{\part y^2}=0$