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[[File:Illustration of Riemann Mapping Theorem.JPG|Illustration of Riemann Mapping Theorem]]
=Elaboration=
In the above figure, consider ''D''{{su|b=1}} and ''D''{{su|b=2}} as two simply connected regions different from ℂ. The [[Riemann mapping theorem]] provides the existence of ''w=f(z)'' mapping ''D''{{su|b=1}} onto the unit disk and existence of ''w=g(z)'' mapping ''D''{{su|b=2}} onto the unit disk. Thus ''g''{{su|p=-1}}''f'' is a one-one mapping of ''D''{{su|b=1}} onto ''D''{{su|b=2}}.
If we can show that ''g''{{su|p=-1}}, and consequently the composition, is analytic, we then have a conformal mapping of ''D''{{su|b=1}} onto ''D''{{su|b=2}}, proving "any two simply connected regions different from the whole plane ℂ can be mapped conformally onto each other."
==Univalent Function==
We know that a complex function is a multiple valued function. That is, for distinct points ''z''{{su|b=1}}, ''z''{{su|b=2}},... in a ___domain ''D'', they may share a common value, ''f(z{{su|b=1}})''=''f(z{{su|b=2}})''=... But if we restrict a complex function to be injective( one-one ), then we obtain a class of functions, viz, univalent functions. A function ''f'' analytic in a ___domain ''D'' is said to be univalent there if it does not take the same value twice for all pairs of distinct points ''z''{{su|b=1}} and ''z''{{su|b=2}} in ''D'', i.e ''f(z{{su|b=1}})''≠''f(z{{su|b=2}})'' implies ''z''{{su|b=1}}≠''z''{{su|b=2}}. Alternate terms in common use are ''schilicht'' and ''simple''. It is a remarkable fact, fundamental to the theory of univalent functions, that univalence is essentially preserved under uniform convergence.
==References==
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