Conformal Mapping of a Parabola onto a Half-Plane
Theorem 1
A conformal mapping maps parabolas to half-planes.
Explanation
Considering what we learned from , 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 again will do the trick.
Proof
Given , and since , then , therefore, represents a parabola in the -plane, and by is mapped to the line in the -plane.
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Osborne (1999). Complex variables and their applications: p214. ↩︎