## Central idea

What conformal mappings are there from the unit disk, $\mathbb{D}, \textrm{ to D a subset of } \mathbb{C}$?

Alternate symbols. Unit disk, $\mathbb{D} \; , \textrm{ or } B_1(0)$

## Slide 3 Riemann Mapping Theorem

D is a simply connected domain in $\mathbb{C}$, but not all of $\mathbb{C}$. D can be a very “irregular ” shape (See diagram) but does not contain any holes.

There is a conformal map that maps D onto the open unit disk $\mathbb{D}$.

Consequence, there is some point $z_0$ in D that maps uniquely onto the origin in $\mathbb{D}$

## Slide 5 Mapping upper half plane to unit disk using a Mobius transformation

D is upper half plane, so contain following points on real line: 0, 1, ∞.

Therefore a (“reverse”) mobius transformation can take these points to points on perimeter on unit circle, $\mathbb{D}$, in same order

f(0) = 1

f(1) = i

f(∞) = -1