Riemann mapping theorem

Every simply connected proper open subset of the complex plane is conformally equivalent to the open unit disk. Once one image point and derivative direction are fixed, the conformal map is unique.

Holomorphic function: A complex-valued function that is complex differentiable at every point of an open domain. The theorem's conformal maps are holomorphic functions with nonzero derivatives.
Simply connected space: A path-connected topological space in which every loop can be continuously contracted to a point. Simple connectivity is the topological hypothesis that rules out holes in the domain.
Bernhard Riemann: A German mathematician whose work shaped complex analysis, geometry, and number theory. His 1851 dissertation introduced the theorem and its influential existence argument.
Conformal welding: A construction that joins two Riemann surfaces by identifying their boundary points through a homeomorphism. Mapping domains to disks provides coordinates for studying how their boundaries can be glued.
Annulus: A planar region between two concentric circles, with the inner disk removed. An annulus is not simply connected, so it cannot be conformally equivalent to the disk.
Conformal map: A differentiable map that preserves angles locally and has a differentiable inverse. The theorem asserts the existence of a conformal map from each eligible domain to the disk.
Open set: A subset of a topological space in which every point has a neighborhood contained in the subset. The theorem concerns domains that are open subsets of the complex plane.
Riemann's 1851 dissertation: Bernhard Riemann's doctoral work on the foundations of complex function theory. It contains the original formulation and proof approach for the mapping theorem.
Schwarz–Christoffel mapping: A formula for conformally mapping the upper half-plane or unit disk onto polygonal domains. It gives explicit maps for important domains covered abstractly by the theorem.
Multiply connected domain: A connected domain whose loops include ones that cannot be contracted to a point within the domain. The theorem excludes these domains because their holes obstruct equivalence with the disk.

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