Conformal Mappings

Conformal Mappings

Definition

Conformality Criterion

Möbius Transformations

Properties of Möbius Transformations

Standard Möbius Transformations

1. Translation:
2. Rotation:
3. Dilation: ()
4. Inversion:

Riemann Mapping Theorem

Schwarz-Christoffel Formula

Maps the upper half-plane to a polygon.

Examples

Example 1: is conformal everywhere except .

Example 2: maps horizontal lines to rays and vertical lines to circles.

Example 3: Cayley transform maps the upper half-plane to the unit disk.

Applications

Harmonic Functions and Conformal Maps

If is harmonic on a domain and is conformal, then is harmonic on .

This allows us to transfer harmonic functions between domains.

Fixed Points

A Möbius transformation (not the identity) has either one or two fixed points.

Automorphisms of the Unit Disk

These are all the conformal self-maps of the unit disk.

Schwarz Lemma

if is holomorphic with .