Lecture 23: Schwarz lemma and introduction to Mobius transformations

We introduce Mobius transformations - otherwise known as fractional linear transforms - by studying the transform that takes the half space of the upper half of the complex plane to the unit disk and studying its inverse. We show the real line is wrapped around the unit circle with infinity being sent to -1 and 0 sent to 1. We then (at 24:00) state and prove Schwartz's lemma. Finally at 51:45 we start to discuss general Mobius transforms and we show any Mobius transform is the composition an affine mapping, the inversion map given by 1/z, and another affine mapping. In other words, after affine change of variable in image and the domain - the the mobius mapping is just the inversion mapping. We then state the theorem that lines and circles are sent to lines and circles by Mobius transformations and due the previous result is enough to show that for the inversion map. We start the proof of this showing that the inversion maps sends circles that intersect the origin to lines.

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