Math 2.0

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

Unit disc

Description

The open unit disc is 𝔻={z∈ℂ:∣z∣<1}\mathbb{D} = \set{z \in \C : \abs{z} < 1}, and the closed unit disc is 𝔻‾={z∈ℂ:∣z∣≤1}\overline{\mathbb{D}} = \set{z \in \C : \abs{z} \le 1}. The open disc is open, since it is the preimage of (−∞,1)(-\infty, 1) under the continuous map z↦∣z∣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 00, so every closed curve in it contracts to 00 along straight segments, and it is a simply connected domain. Its boundary is the unit circle 𝕋={z∈ℂ:∣z∣=1}\mathbb{T} = \set{z \in \C : \abs{z} = 1}.

Remarks

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

Where this leads

30 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Unit_disk
About this page
Tag8G5ABXRK85
URIm2://8G5ABXRK85
Kindobject
WritingW0 stub
MSC30J99, 30C20
Instance ofSimply connected domain, Domain in the complex plane
Same as
Changed2 November 2026

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