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Conformal Mapping of a Parabola onto a Half-Plane 📂Complex Anaylsis

Conformal Mapping of a Parabola onto a Half-Plane

Theorem 1

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A conformal mapping w=f(z)=z1/2\displaystyle w = f(z) = z^{1/2} maps parabolas to half-planes.

Explanation

Considering what we learned from R2\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 ξ=wa\xi = w - a again will do the trick.

Proof

Given z=x+iyw=u+iv z = x + i y \\ w = u + i v , and since z=w2=(u+iv)2=u2v2+i2uv=x+iy z = w^2 = (u + iv)^2 = u^2 - v^2 + i 2 uv = x + iy , then 4u2(u2x)=y2 4 u^2 (u^2 - x ) = y^2 , therefore, y2=4a2(a2x)y^2 = 4 a^2 (a^2 - x ) represents a parabola in the ZZ-plane, and by ff is mapped to the line WW in the u=au=a-plane.


  1. Osborne (1999). Complex variables and their applications: p214. ↩︎