Past exam of the metric spaces and complex analysis course of the University of Oxford / 2023 / 6 / a

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.

At $z_0$, complex differentiability givesWriting $f^{\prime }(z_0)=\lambda e^{i\alpha }$, the assumption $f^{\prime }(z_0)\ne 0$ gives $\lambda >0$. Its real derivative is therefore the similarity $h\mapsto \lambda e^{i\alpha }h$, a rotation through $\alpha$ followed by scaling by $\lambda$.

More explicitly, for two differentiable curves ${\gamma }_1,{\gamma }_2$ with ${\gamma }_j(0)=z_0$ and nonzero tangent vectors, the chain rule givesTaking the complex argument of this ratio shows that the oriented angle is preserved, while nonvanishing ensures both image tangents remain regular. Thus