Functions of a complex variable Function spaces and the unit disc Object

Unit disc

Description

The open unit disc is $\mathbb{D} = \set{z \in \C : \abs{z} < 1}$ , and the closed unit disc is $\overline{\mathbb{D}} = \set{z \in \C : \abs{z} \le 1}$ . The open disc is open, since it is the preimage of $(-\infty, 1)$ under the continuous map $z \mapsto \abs{z}$ , and it is connected, since it is convex. So it is a domain in the complex plane. It is star-shaped about $0$ , so every closed curve in it contracts to $0$ along straight segments, and it is a simply connected domain. Its boundary is the unit circle $\mathbb{T} = \set{z \in \C : \abs{z} = 1}$ .

Remarks

By the Riemann mapping theorem, every simply connected domain other than $\C$ is conformally equivalent to $\mathbb{D}$ . The Cayley map $z \mapsto (z - i)/(z + i)$ is a conformal map of the upper half-plane onto $\mathbb{D}$ . The conformal automorphisms of $\mathbb{D}$ are the Möbius transformations $z \mapsto e^{i\theta}(z - a)/(1 - \bar a z)$ with $\abs{a} < 1$ and real $\theta$ . With the hyperbolic metric it becomes the Poincaré disk model of the hyperbolic plane.

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Tag	`8G5ABXRK85`
URI	`m2://8G5ABXRK85`
Kind	object
Writing	W0 stub
MSC	30J99, 30C20
Instance of	Simply connected domain, Domain in the complex plane
Same as	Wikidata Q2390323 Wikipedia Unit disk

Unit disc

Description

Remarks

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