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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If ''K'' is 0, then the function is [[conformal map|conformal]].

===Analytic continuation===

[[Image:Imaginary log analytic continuation.png|316px|right|thumb|Analytic continuation of natural logarithm (imaginary part)]]

'''Analytic continuation''' is a technique to extend the [[___domain of a function|___domain]] of a given [[analytic function]]. Analytic continuation often succeeds in defining further values of a function, for example in a new region where an [[infinite series]] representation in terms of which it is initially defined becomes divergent.

The step-wise continuation technique may, however, come up against difficulties. These may have an essentially topological nature, leading to inconsistencies (defining more than one value). They may alternatively have to do with the presence of [[mathematical singularities]]. The case of [[several complex variables]] is rather different, since singularities then cannot be isolated points, and its investigation was a major reason for the development of [[sheaf cohomology]].

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===Extremal problems===

Topics in this area include "Maximum principle; Schwarz's lemma, Lindelöf principle, analogues and generalizations" .<ref>MSC80 in the MSC classification system</ref>.

===Univalent and multivalent functions===

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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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If we can show that <math>g^{-1}</math>, and consequently the composition, is analytic, we then have a conformal mapping of <math>D_1</math> onto <math>D_2</math>, proving "any two simply connected regions different from the whole plane <math>\mathbb C</math> can be mapped conformally onto each other."

===SchwartzSchwarz's Lemma===

{{main|Schwartz'sSchwarz Lemmalemma}}

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===

{{main|Maximum principle}}

The [[maximum principle]] is a property of solutions to certain [[partial differential equations]], of the [[elliptic partial differential equation|elliptic]] and [[parabolic partial differential equation|parabolic]] types. Roughly speaking, it says that the [[maximum]] of a function in a [[Domain (mathematical analysis)|___domain]] is to be found on the boundary of that ___domain. Specifically, the ''strong'' maximum principle says that if a function achieves its maximum in the interior of the ___domain, the function is uniformly a constant. The ''weak'' maximum principle says that the maximum of the function is to be found on the boundary, but may re-occur in the interior as well. Other, even weaker maximum principles exist which merely bound a function in terms of its maximum on the boundary.

===Riemann-Hurwitz formula===

{{main|Riemann-Hurwitz formula}}

the '''Riemann–Hurwitz formula''', named after [[Bernhard Riemann]] and [[Adolf Hurwitz]], describes the relationship of the [[Euler characteristic]]s of two [[Surface (topology)|surface]]s when one is a ''ramified covering'' of the other. It therefore connects [[Ramification (mathematics)|ramification]] with [[algebraic topology]], in this case. It is a prototype result for many others, and is often applied in the theory of [[Riemann surface]]s (which is its origin) and [[algebraic curve]]s.

==== Statement ====

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That is because each simplex of ''S'' should be covered by exactly ''N'' in ''S''&prime; &mdash; at least if we use a fine enough [[Triangulation (geometry)|triangulation]] of ''S'', as we are entitled to do since the Euler characteristic is a [[topological invariant]]. What the Riemann–Hurwitz formula does is to add in a correction to allow for ramification (''sheets coming together'').

Now assume that ''S'' and ''S&prime;'' are [[Riemann surface]]s, and that the map π is [[analytic function|complex analytic]]. The map π is said to be ''ramified'' at a point ''P'' in ''S''&prime; if there exist analytic coordinates near ''P'' and π(''P'') such that π takes the form π(''z'') = ''z''^''n'', and ''n''&nbsp;>&nbsp;1. An equivalent way of thinking about this is that there exists a small neighborhood ''U'' of ''P'' such that π(''P'') has exactly one preimage in ''U'', but the image of any other point in ''U'' has exactly ''n'' preimages in ''U''. The number ''n'' is called the ''[[ramification index]] at P'' and also denoted by ''e''_''P''. In calculating the Euler characteristic of ''S''&prime; we notice the loss of ''e_P''&nbsp;&minus;&nbsp;1 copies of ''P'' above π(''P'') (that is, in the inverse image of π(''P'')). Now let us choose triangulations of ''S'' and ''S&prime;'' with vertices at the branch and ramification points, respectively, and use these to compute the Euler characteristics. Then ''S&prime;'' will have the same number of ''d''-dimensional faces for ''d'' different from zero, but fewer than expected vertices. Therefore, we find a "corrected" formula

:<math>\chi(S') = N\cdot\chi(S) - \sum_{P\in S'} (e_P -1) </math>

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==References==

{{Reflist}}

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

* {{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}}.

*{{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 = 1–2 | year = 2012 | url = http://downloads.hindawi.com/journals/ijmms/2010/906317.pdf | doi-access = free }}

*{{cite book |title=Lecture notes on Introduction to Univalent Functions|

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

first=K.I|last=Noor|publisher=CIIT, Islamabad, Pakistan}}

*{{Cite doi|10.1155/2012/976374}}

*{{Cite isbn | 9780821852705}}

[[Category:Analytic functions|*]]