The Riemann Mapping theorem is one of the most useful theorems in elementary complex analysis. From a planar topology viewpoint we know that there exists simply connected domains with complicated boundaries [1, 3, 5]. For such domains there are no obvious homeomorphisms between them. However the Riemann mapping theorem states that such simply connected domains are not only homeomorphic but also biholomorphic

Walsh [6] presented History of the Riemann Mapping Theorem, as an outline of how proofs of the Riemann Mapping theorem have evolved over time. A very important theorem in Complex Analysis, Riemann’s mapping theorem was first stated, with an incorrect proof, by Bernhard Riemann in his inaugural dissertation in 1851. Since the publication of this “proof” various objections have been made which all in all lead to this proof being labeled as incorrect. While Riemann’s proof is incorrect it did provide the general guidelines, via the Dirichlet Principle and Green’s function, which would prove vital in future proofs.

It would not be until 1900 that the American mathematician Osgood produced a valid proof of the theorem. Osgood’s proof utilized the original ideas of Riemann but was made more difficult than today’s methods of proof as he did not possess Perron’s method of solving the Dirichlet problem [4]. Rather he used approximations from the interior of a simply connected region and took limits of the piecewise linear case of the Dirichlet problem which had been solved by Schwarz years earlier. Although this is considered the first correct proof, Osgood did not receive recognition for his achievement with many mathematicians such as Koebe receiving more recognition for giving a proof years later. As a consequence of many years of study many other successful proofs have been presented over the years using various methods. Such being the proof by Hilbert using the Calculus of Variations and the proof by F. Riesz and L. Fejer using Montel’s theory of normal families of functions.