Conformal map

A conformal map is a function between domains in the complex plane that preserves angles locally: the intersection angle between any two curves at a point is the same as the angle between their images under the map. In complex analysis, a map is conformal if and only if it is holomorphic with non-vanishing derivative, which means that conformal mappings are precisely the analytic functions that do not fold or tear the plane. This equivalence between geometric preservation and analytic regularity is one of the central miracles of complex function theory.

The practical importance of conformal mapping lies in its capacity to transform problems from geometrically complicated domains into simpler ones. The Riemann mapping theorem guarantees that any simply connected proper subset of the complex plane can be mapped conformally onto the unit disk, though the theorem offers no prescription for finding the map. For specific applications — mapping airfoils to circles, or polygons to half-planes — explicit constructions such as the Joukowsky transform and the Schwarz-Christoffel mapping provide the necessary tools. Conformal maps also appear in physics wherever harmonic functions govern a field: electrostatics, heat conduction, and ideal fluid flow all reduce to solving Laplace's equation in a domain that can be simplified through conformal transformation.