📂Complex Anaylsis

Mapping a Trapezoid to a Circle through Conformal Mapping

Theorem ¹

A conformal mapping $\displaystyle w = f(z) = z^{n}$ maps a sector to a semicircle.

Explanation

If we consider the radius of the sector to be infinite, then $f$ can be thought of as mapping an angle to a straight angle and its interior to a half-plane.

Similarly, since a semicircle is also a sector and a half-plane is also an angle, by applying $\xi = w^{2}$ once more, it can be mapped to a complete circle or plane.