📂Complex Anaylsis

Conformal Mapping of a Parabola onto a Half-Plane

Theorem ¹

A conformal mapping $\displaystyle w = f(z) = z^{1/2}$ maps parabolas to half-planes.

Explanation

Considering what we learned from $\mathbb{R}^2$, it might seem obvious, but it’s necessary to check whether this holds in the complex plane as well. If you want to cleanly divide it along the y-axis, taking $\xi = w - a$ again will do the trick.

Proof

Given $$ z = x + i y \\ w = u + i v $$, and since $$ z = w^2 = (u + iv)^2 = u^2 - v^2 + i 2 uv = x + iy $$, then $$ 4 u^2 (u^2 - x ) = y^2 $$, therefore, $y^2 = 4 a^2 (a^2 - x )$ represents a parabola in the $Z$-plane, and by $f$ is mapped to the line $W$ in the $u=a$-plane.

■