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 $z(x+iy)=u(x,y)+iv(x,y)$ , the Cauchy–Riemann conditions are

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

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

${\frac {\partial u}{\partial r}}={\frac {1}{r}}{\frac {\partial v}{\partial \theta }}\ ,\ {\frac {\partial v}{\partial r}}=-{\frac {1}{r}}{\frac {\partial u}{\partial \theta }}$

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.

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