Editing Unit disk (section)

== The open unit disk, the plane, and the upper half-plane ==
The function

:<math>f(z)=\frac{z}{1-|z|^2}</math>

is an example of a real [[analytic function|analytic]] and [[bijective]] function from the open unit disk to the plane; its inverse function is also analytic. Considered as a real 2-dimensional [[analytic manifold]], the open unit disk is therefore isomorphic to the whole plane. In particular, the open unit disk is [[homeomorphic]] to the whole plane.

There is however no [[conformal map|conformal]] bijective map between the open unit disk and the plane. Considered as a [[Riemann surface]], the open unit disk is therefore different from the [[complex plane]].

There are conformal bijective maps between the open unit disk and the open [[upper half-plane]]. So considered as a Riemann surface, the open unit disk is isomorphic ("biholomorphic", or "conformally equivalent") to the upper half-plane, and the two are often used interchangeably.

Much more generally, the [[Riemann mapping theorem]] states that every [[simply connected]] [[open set|open subset]] of the complex plane that is different from the complex plane itself admits a conformal and bijective map to the open unit disk.

One bijective conformal map from the open unit disk to the open upper half-plane is the [[Möbius transformation]]

:<math>g(z)=i\frac{1+z}{1-z}</math> &nbsp; which is the inverse of the [[Cayley transform#Complex homography|Cayley transform]].

Geometrically, one can imagine the real axis being bent and shrunk so that the upper half-plane becomes the disk's interior and the real axis forms the disk's circumference, save for one point at the top, the "point at infinity". A bijective conformal map from the open unit disk to the open upper half-plane can also be constructed as the composition of two [[stereographic projection]]s: first the unit disk is stereographically projected upward onto the unit upper half-sphere, taking the "south-pole" of the unit sphere as the projection center, and then this half-sphere is projected sideways onto a vertical half-plane touching the sphere, taking the point on the half-sphere opposite to the touching point as projection center.

The unit disk and the upper half-plane are not interchangeable as domains for [[Hardy spaces]]. Contributing to this difference is the fact that the unit circle has finite (one-dimensional) [[Lebesgue measure]] while the real line does not.