Let ''z''{{su|b=0}}<math>z_0</math> be a point in a simply-connected region ''D''{{su|b=1}}<math>D_1 (''D''D_1 \neq \mathbb{{su|b=1C}}≠ ℂ)</math> and ''D''{{su|b=1}}<math>D_1</math> having at least two boundary points. Then there exists a unique analytic function ''<math>w=f(z)''</math> mapping ''D''{{su|b=1}}<math>D_1</math> bijectively into the open unit disk <math>|''w''| < 1</math> such that ''<math>f(''z''{{su|b=0}}z_0)''=0</math> and <math>f'(z_0) > 0</math>.
''f ′(''z''{{su|b=0}})''>0.
It should be noted that while [[Riemann's mapping theorem]] demonstrates the existence of a mapping function, it does not actually ''exhibit'' this function.
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===Elaboration===
In the above figure, consider ''D''{{su|b=1}}<math>D_1</math> and ''D''{{su|b=2}}<math>D_2</math> as two simply connected regions different from ℂ<math>\mathbb C</math>. The [[Riemann mapping theorem]] provides the existence of ''<math>w=f(z)''</math> mapping ''D''{{su|b=1}}<math>D_1</math> onto the unit disk and existence of ''<math>w=g(z)''</math> mapping ''D''{{su|b=2}}<math>D_2</math> onto the unit disk. Thus ''<math>g''^{{su|p=-1}}''f''</math> is a one-to-one mapping of ''D''{{su|b=1}}<math>D_1</math> onto ''D''{{su|b=2}}<math>D_2</math>.
If we can show that ''<math>g''^{{su|p=-1}}</math>, and consequently the composition, is analytic, we then have a conformal mapping of ''D''{{su|b=1}}<math>D_1</math> onto ''D''{{su|b=2}}<math>D_2</math>, proving "any two simply connected regions different from the whole plane ℂ<math>\mathbb C</math> can be mapped conformally onto each other."
==Univalent function==
Of special interest are those complex functions which are one-to-one. That is, for points ''z''{{su|b=1}}<math>z_1</math>, ''z''{{su|b=2}}<math>z_2</math>, in a ___domain ''<math>D''</math>, they share a common value, ''<math>f(z{{su|b=1}}z_1)''=''f(z{{su|b=2}}z_2)''</math> only if they are the same point (''z''{{su|b<math>z_1=1}} = ''z''{{su|b=2}}'')z_2</math>. A function ''<math>f''</math> analytic in a ___domain ''<math>D''</math> 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}}<math>z_1</math> and ''z''{{su|b=2}}<math>z_2</math> in ''<math>D''</math>, i.e ''<math>f(z{{su|b=1}}z_1)''≠'' \neq f(z{{su|b=2}}z_2)''</math> implies ''z''{{su|b=1}}≠''z''{{su|b=2}}<math>z_1 \neq z_2</math>. 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.
