Riemann Mapping Theorem
Let z_0 be a point in a simply connected region R!=C, where C is the complex plane. Then there is a unique analytic function w=f(z) mapping R one-to-one onto the disk |w|<1 such that f(z_0)=0 and f^'(z_0)>0. The corollary guarantees that any two simply connected regions except R^2 (the Euclidean plane) can be mapped conformally onto each other.
See also
Conformal MappingExplore with Wolfram|Alpha
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References
Krantz, S. G. "The Riemann Mapping Theorem." §6.4 in Handbook of Complex Variables. Boston, MA: Birkhäuser, pp. 86-87, 1999.Referenced on Wolfram|Alpha
Riemann Mapping TheoremCite this as:
Weisstein, Eric W. "Riemann Mapping Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/RiemannMappingTheorem.html