Unit Disk - The Open Unit Disk, The Plane, and The Upper Half-plane

The Open Unit Disk, The Plane, and The Upper Half-plane

The function

is an example of a real 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 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 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

which is the inverse of the 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 projections: 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.