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'''Geometric function theory''' is the study of [[Geometry|geometric]] properties of [[analytic function]]s. A fundamental result in the theory is the [[Riemann mapping theorem]].
==Riemann
Let ''z''{{su|b=0}} be a point in a simply-connected region ''D''{{su|b=1}} (''D''{{su|b=1}}≠ ℂ) and ''D''{{su|b=1}} having at least two boundary points. Then there exists a unique analytic function ''w = f(z)'' mapping ''D''{{su|b=1}} bijectively into the open unit disk |''w''|<1 such that ''f(''z''{{su|b=0}})''=0 and
''Re 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.
[[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
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==
*{{cite book |title=Geometric Function Theory: Explorations in Complex Analysis|
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*{{cite book |title=Lecture notes on Introduction to Univalent Functions|
first=K.I|last=Noor|publisher=CIIT, Islamabad, Pakistan}}
{{mathanalysis-stub}}
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