In this topic we will look at the geometric notion of conformal maps. It will turn out that analytic functions are automatically conformal. Once we have understood the general notion, we will look at a specific family of conformal maps called fractional linear transformations and, in particular at their geometric properties. As an application we will use fractional linear transformations to solve the Dirichlet problem for harmonic functions on the unit disk with specified values on the unit circle. At the end we will return to some questions of fluid flow.
- 11.1: Geometric Definition of Conformal Mappings
- Conformal maps are functions on C that preserve the angles between curves.
- 11.5: Riemann Mapping Theorem
- The Riemann mapping theorem is a major theorem on conformal maps.
Thumbnail: A rectangular grid under a conformal map. It is seen that maps pairs of lines intersecting at 90° to pairs of curves still intersecting at 90°. (Public Domain; Oleg Alexandrov via Wikipedia)