Geometric function theory: Difference between revisions

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{{Short description|Study of space and shapes locally given by a convergent power series}}

'''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]].

==Topics in geometric function theory==

The following are some of the most important topics in geometric function theory:<ref>Hurwitz-Courant, ''Vorlesunger über allgemeine Funcktionen Theorie'', 1922 (4th ed., appendix by H. Röhrl, vol. 3, ''Grundlehren der mathematischen Wissenschaften''. Springer, 1964.)</ref><ref>MSC classification for 30CXX, Geometric Function Theory, retrieved from http://www.ams.org/msc/msc2010.html on September 16, 2014.</ref>

===Conformal maps===

{{main|Conformal map}}

[[Image:Conformal map.svg|right|thumb|A rectangular grid (top) and its image under a conformal map ''f'' (bottom). It is seen that ''f'' maps pairs of lines intersecting at 90&deg;° to pairs of curves still intersecting at 90&deg;°.]]

A '''conformal map''' is a [[function (mathematics)|function]] which preserves [[angle]]s locally. In the most common case the function has a [[Domain of a function|___domain]] and [[Range (mathematics)of a function|range]] in the [[complex plane]].

More formally, a map,

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Let <math>z_0</math> be a point in a simply-connected region <math>D_1 (D_1 \neq \mathbb{C})</math> and <math>D_1</math> having at least two boundary points. Then there exists a unique analytic function <math>w=f(z)</math> mapping <math>D_1</math> bijectively into the open unit disk <math>|w| < 1</math> such that <math>f(z_0)=0</math> and <math>f'(z_0) > 0</math>.

It should be noted that whileAlthough [[Riemann's mapping theorem]] demonstrates the existence of a mapping function, it does not actually ''exhibit'' this function. An example is given below.

[[File:Illustration of Riemann Mapping Theorem.JPG|Illustration of Riemann Mapping Theorem]]

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The '''Schwarz lemma''', named after [[Hermann Amandus Schwarz]], is a result in [[complex analysis]] about [[holomorphic functions]] from the [[open set|open]] [[unit disk]] to itself. The lemma is less celebrated than stronger theorems, such as the [[Riemann mapping theorem]], which it helps to prove. It is however one of the simplest results capturing the rigidity of holomorphic functions.

====Statement====

<blockquote>'''Schwarz Lemma.''' Let '''D''' = {''z'' : |''z''| < 1} be the open [[unit disk]] in the [[complex number|complex plane]] '''C''' centered at the [[origin (mathematics)|origin]] and let ''f'' : '''D''' → '''D''' be a [[holomorphic map]] such that ''f''(0) = 0.

Then, |''f''(''z'')| ≤ |''z''| for all ''z'' in '''D''' and |''f′''(0)| ≤ 1.

Moreover, if |''f''(''z'')| = |''z''| for some non-zero ''z'' or if |''f′''(0)| = 1, then ''f''(''z'') = ''az'' for some ''a'' in '''C''' with |''a''| =(necessarily) equal to 1.</blockquote>

===Maximum principle===

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==References==

{{Reflist}}

* {{Citation |last=Ahlfors |first=Lars |author-link=Lars Ahlfors |title=Zur Theorie der Überlagerungsflächen |journal=[[Acta Mathematica]] |volume=65 |issue=1 |pages=157–194 |year=1935 |issn=0001-5962 |language=de |doi=10.1007/BF02420945 |jfm=61.0365.03 |zbl=0012.17204 |doi-access=free}}.

* Hurwitz-Courant, ''Vorlesunger über allgemeine Funcktionen Theorie'', 1922 (4th ed.,appendix by H. Röhrl, vol. 3, ''Grundlehren der mathematischen Wissenschaften''. Springer, 1964.)

*{{Citation |last=Grötzsch |first=Herbert |author-link=Herbert Grötzsch |title=Über einige Extremalprobleme der konformen Abbildung. I, II. |language=de |year=1928 |journal=[[Berichte über die Verhandlungen der Königlich Sächsischen Gesellschaft der Wissenschaften zu Leipzig. Mathematisch-Physische Classe]] |volume=80 |pages=367–376, 497–502 |jfm=54.0378.01}}.

* Hurwitz-Courant, ''Vorlesunger über allgemeine Funcktionen Theorie'', 1922 (4th ed., appendix by H. Röhrl, vol. 3, ''Grundlehren der mathematischen Wissenschaften''. Springer, 1964.)

*{{cite book |title=Geometric Function Theory: Explorations in Complex Analysis|

first=Steven|last=Krantz|publisher=Springer|year=2006|isbn=0-8176-4339-7}}

*{{Cite journal | last1 = Bulboacă | first1 = T. | last2 = Cho | first2 = N. E. | last3 = Kanas | first3 = S. A. R. | title = New Trends in Geometric Function Theory 2011 | doi = 10.1155/2012/976374 | journal = International Journal of Mathematics and Mathematical Sciences | volume = 2012 | pages = 11–2 | year = 2012 | pmidurl = http://downloads.hindawi.com/journals/ijmms/2010/906317.pdf | pmcdoi-access = free }}

*{{cite book | isbn = 978-0821852705 | title = Conformal Invariants: Topics in Geometric Function Theory | last1 = Ahlfors | first1 = Lars | year = 2010 | publisher = AMS Chelsea Publishing | pages = }}

[[Category:Analytic functions|*]]