The Riemann mapping theorem is the most celebrated result of complex analysis. It is one of the most important results of 19th century mathematics. In modern times, it is the beginning of the study of complex analysis from a geometric viewpoint.

Riemann correctly stated the theorem, but unfortunately his proof of the theorem was lacking. According to various accounts (I've not been able to form a coherent account, but this may be due to the difficulties of establishing what precisely had been proven in the late 19th century), he assumed but did not prove that a certain maximal problem had a solution, or alternatively that a certain Dirichelet boundary value problem had a solution. A final proof was definitely known by the early 20th century; different sources attribute it to Koebe, Osgood, or others.

Theorem. Let U⊂C be a simply connected open domain which is not the entire plane of complex numbers C, and let D be the unit disk. Let a∈U be a point in U. Then there exists a unique holomorphic mapping f: U→D such that:

An f as in the Riemann mapping theorem is sometimes called a conformal mapping. "Conformal" here means that f preserves angles locally: the image of 2 arcs through any point preserves the angles between them.

A note on U: since U is simply connected and not C, it follows that U must omit some point x∈C\U, and that this x must be connected by a path avoiding U to infinity (e.g. on the Riemann sphere). This is often given as an alternative characterisation of U.

Notes on the properties of f: The "meat" of the theorem is in the existence of f:U→D which is one-to-one and onto. The additional 2 properties are added to guarantee uniqueness. Without f'(a)>0 (and in particular, real), we could compose f with any rotation to get another f. And without f(a)=0, we could compose f with some Mobius function which maps another point of the disk to 0. Indeed, uniqueness is the easy part of the proof -- it follows almost directly from the Schwarz lemma.

Note well what this theorem is saying! If we take U to be some open square, then we are guaranteed a holomorphic function f which maps the inside of a square to the inside of a circle! (This f also preserves angles -- so the distortion near a corner of the square will be tremendous). Today, such an f can be constructed using elliptic functions -- but using the Riemann mapping theorem is much easier.

Some constructions are easier to find explicitly. For instance, if U is the lower half plane {z: Im(z) < 0}, then the mapping f(z)=(z+i)/(iz+1) maps U onto C. Other Mobius functions let you map any other disk or half plane onto the unit disk.

If V=C\{x+0i: x∈R and x<0} is the complex plane minus the ray of negative numbers, then a branch of the square root function exists mapping V to the half plane U above -- which we've seen how to map to the unit disk. "Strips" of the complex plane can similarly be transformed by using the exponent function. A favourite of Math TAs is to ask Physics undergraduates to transform various domains onto the unit disk -- Riemann says it's possible, but with a very non-constructive proof.

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