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 , the Cauchy-Riemann conditions are
or, using the polar representation of a complex function in terms of
For any function which respects the Cauchy-Riemann conditions, will also obey Laplace's equation. This can easily be seen by differentiating a second time.