The bookUnit 2 · Zeros and Mappings37 / 83

Conformal maps, Möbius transformations, Schwarz lemma

Why this is asked: Know the dictionary of standard maps between disc, half-plane, strip and sector, and that Möbius maps send circles-and-lines to circles-and-lines and preserve cross-ratio.

Conformal maps and Möbius transformations

The written notes for this page come with the Notes pack. The video, the visual and the practice below are free.

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See it move

Möbius maps bending the gridinteractive

Slide the map on and watch straight lines bow into circles — every one still a circle or a line, angles preserved throughout.

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The trap here

“ $\mathbb{C}$ and the unit disc are biholomorphic (both are simply connected)” — false

Liouville's theorem

A biholomorphism 𝔻 $\to \mathbb{C}$ would invert to a bounded entire function. $\mathbb{C}$ is the sole exception in the Riemann mapping theorem.