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 $f:ℂ\mapsto ℂ$ 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 $g:ℂ\mapsto ℂ$ 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
Canonical name	ExampleOfConformalMapping
Date of creation	2013-03-22 13:36:36
Last modified on	2013-03-22 13:36:36
Owner	Johan (1032)
Last modified by	Johan (1032)
Numerical id	6
Author	Johan (1032)
Entry type	Example
Classification	msc 30E20
Related topic	CategoryOfRiemannianManifolds