📂Complex Anaylsis

Conformal Mapping That Maps a Parabola to a Half-Plane

Theorem ¹

The conformal mapping $\displaystyle w = f(z) = z^{1/2}$ maps a parabola to a half-plane.

Explanation

Recalling what we learned in $\mathbb{R}^2$, this is fairly obvious, but we still need to check whether it also holds in the complex plane. If we want to cleanly split along the vertical axis, we just take $\xi = w - a$ one more time.

Proof

$$ z = x + i y \\ w = u + i v $$ Setting the above, since $$ z = w^2 = (u + iv)^2 = u^2 - v^2 + i 2 uv = x + iy $$ we have $$ 4 u^2 (u^2 - x ) = y^2 $$ Therefore, $y^2 = 4 a^2 (a^2 - x )$ is a parabola in the $Z$-plane, and by $f$ it corresponds to the line $u=a$ on the $W$-plane.

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