# Riemann theorem

Given any two simply-connected domains $G _ {1}$ and $G _ {2}$ of the extended complex plane $\overline{\mathbf C}\;$, distinct from $\overline{\mathbf C}\;$ and also from $\overline{\mathbf C}\;$ with a point excluded from it, then an infinite number of analytic single-valued functions on $G _ {1}$ can be found such that each one realizes a one-to-one conformal transformation of $G _ {1}$ onto $G _ {2}$. In this case, for any pair of points $a \in G _ {1}$, $a \neq \infty$, and $b \in G _ {2}$ and any real number $\alpha$, $0 \leq \alpha \leq 2 \pi$, a unique function $f$ of this class can be found for which $f( a) = b$, $\mathop{\rm arg} f ^ { \prime } ( a) = \alpha$. The condition $\mathop{\rm arg} f ^ { \prime } ( a) = \alpha$ geometrically means that each infinitely-small vector emanating from the point $a$ changes under the transformation $w = f( z)$ into an infinitely-small vector the direction of which forms with the direction of the original vector the angle $\alpha$.
Riemann's theorem is fundamental in the theory of conformal mapping and in the geometrical theory of functions of a complex variable in general. In addition to its generalizations to multiply-connected domains, it finds wide application in the theory of functions of a complex variable, in mathematical physics, in the theory of elasticity, in aero- and hydromechanics, in electro- and magnetostatics, etc. This theorem was formulated by B. Riemann (1851) for the more general case of simply-connected and, generally speaking, non-single sheeted domains over the complex plane. Instead of using the normalizing conditions "fa= b, argf'a=a" of the conformal mapping $w = f( z)$, which guarantee its uniqueness, Riemann used for the same purpose the conditions "fa= b, fz=w" , where $a \in G _ {1}$, $b \in G _ {2}$ and $\zeta$ and $\omega$ are points of the boundaries of $G _ {1}$ and $G _ {2}$, respectively, given in advance. The last conditions are not always correct, given the contemporary definition of a simply-connected domain. In proving his theorem, Riemann drew to a considerable degree on concepts of physics, which also convinced him of the importance of this theorem for applications. D. Hilbert made Riemann's proof mathematically precise by substantiating the so-called Dirichlet principle, which was used by Riemann in his proof.