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.
Read more about this topic: Unit Disk
Famous quotes containing the words the open, open, unit and/or upper:
“The moment when she crawled out onto the back of the open limousine in which her husband had been murdered was the first and last time the American people would see Jacqueline Bouvier Kennedy Onassis crawl.... She was the last great private public figure in this country. In a time of gilt and glitz and perpetual revelation, she was perpetually associated with that thing so difficult to describe yet so simple to recognize, the apotheosis of dignity.”
—Anna Quindlen (b. 1952)
“The world is a puzzling place today. All these banks sending us credit cards, with our names on them. Well, we didn’t order any credit cards! We don’t spend what we don’t have. So we just cut them in half and throw them out, just as soon as we open them in the mail. Imagine a bank sending credit cards to two ladies over a hundred years old! What are those folks thinking?”
—Sarah Louise Delany (b. 1889)
“During the Suffragette revolt of 1913 I ... [urged] that what was needed was not the vote, but a constitutional amendment enacting that all representative bodies shall consist of women and men in equal numbers, whether elected or nominated or coopted or registered or picked up in the street like a coroner’s jury. In the case of elected bodies the only way of effecting this is by the Coupled Vote. The representative unit must not be a man or a woman but a man and a woman.”
—George Bernard Shaw (1856–1950)
“The enemy are no match for us in a fair fight.... The young men ... of the upper class are kind-hearted, good-natured fellows, who are unfit as possible for the business they are in. They have courage but no endurance, enterprise, or energy. The lower class are cowardly, cunning, and lazy. The height of their ambition is to shoot a Yankee from some place of safety.”
—Rutherford Birchard Hayes (1822–1893)