Conformal Mapping by Exponential Function
Theorem 1
Conformal mapping $w = f(z) = e^{z} = e^{x} e^{i y}$ maps a rectangle to a sector or an annulus.
Explanation
$f(z) = e^{z}$ is clearly a conformal mapping, but since it is not injective, various restrictions are necessary when considering its inverse mapping.
Osborne (1999). Complex variables and their applications: p217. ↩︎