Conformal Maps Summary

Complex Analysis · Axiom Academy

Section 11 in one view: how angle-preserving maps reshape hard regions into easy ones — and why the geometry survives the trip. A map is conformal exactly where it is analytic with — there it preserves angles, acting locally as a rotation by and a scaling by |f'| . Möbius transformations (with ) are the workhorses: they map circles-and-lines to circles-and-lines, and three source points can be sent to any three targets. The cross-ratio is the one quantity every Möbius map leaves unchanged — it is what lets you pin a transformation down from point data. A small toolkit handles most regions: z^n multiplies angles, e^z turns strips into sectors, and the Cayley map carries the upper half-plane onto the unit disk. The payoff is real: since harmonic functions stay harmonic under a conformal map, a hard boundary-value problem becomes an easy one on a disk or half-plane, then maps back. Core Concept What Makes a Map Conformal At a point where f is analytic and , the map preserves the angle (and the orientation) between any two curves crossing there. Locally it is just multiplication by f'(z_0) : a rotation by and a uniform scaling by |f'(z_0)| . When to use: any time you need to move data between regions without distorting local geometry. Watch out for: points with f'(z_0)=0 are not conformal — angles get multiplied there. Core Concept The Standard Toolkit

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