example of conformal mapping

Consider the four curves $A=\{t\}$, $B=\{t+it\}$, $C=\{it\}$ and $D=\{-t+it\}$, $t\in[-10,10]$. Suppose there is a mapping $f:\mathbb{C}\mapsto\mathbb{C}$ which maps $A$ to $D$ and $B$ to $C$. Is $f$ conformal at $z_{0}=0$? The size of the angles between $A$ and $B$ at the point of intersection $z_{0}=0$ is preserved, however the orientation is not. Therefore $f$ is not conformal at $z_{0}=0$. Now suppose there is a function $g:\mathbb{C}\mapsto\mathbb{C}$ which maps $A$ to $C$ and $B$ to $D$. In this case we see not only that the size of the angles is preserved, but also the orientation. Therefore $g$ is conformal at $z_{0}=0$.

Title example of conformal mapping ExampleOfConformalMapping 2013-03-22 13:36:36 2013-03-22 13:36:36 Johan (1032) Johan (1032) 6 Johan (1032) Example msc 30E20 CategoryOfRiemannianManifolds