 Solution

A conformal map at a point $z_0$ is a map that preserves oriented angles between regular differentiable curves through $z_0$. In the differential formulation, it is real differentiable at $z_0$ and its derivative is an orientation-preserving similarity: a rotation followed by multiplication of lengths by a positive constant. The map is conformal on $U$ when it has this property at every point:Here $R_{\alpha }$ is a planar rotation. Conformality is a local property and does not itself require global injectivity. We use the oriented convention standard in complex analysis; a reflection reverses oriented angles.