Coursera: Analysis of a Complex Kind. 4.5 Riemann Mapping Theorem

Jul 28, 2016

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

slide 2

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

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