Conformal Mapping That Maps a Sector to a Circle
Theorem 1
The conformal mapping $\displaystyle w = f(z) = z^{n}$ maps a sector to a semicircle.
Explanation
If we think of the radius of the sector as being infinite, then $f$ sends the angle to a straight angle and maps its interior to a half-plane.
On the other hand, a semicircle is also a sector and a half-plane is also an angle, so by applying $\xi = w^{2}$ once more we can map it onto a complete circle or plane.
Osborne (1999). Complex variables and their applications: p212. ↩︎