On the complex plane, the set

**D** = {*z*: |*z*| < 1}.

That is, the set of all

points whose

distance to the

origin is

strictly less than 1. Note that we do

*not* include the

unit circle in the unit

disk!

The unit disk is a very common domain in complex analysis and harmonic analysis. It has a compact closure, which makes various questions (such as convergence of a function defined inside the disc to a "nice" function defined on its boundary, the unit circle) interesting.

The Riemann mapping theorem makes the unit circle conformally equivalent to any other simply connected domain, except the entire plane.