Geometric function theory: Difference between revisions

Revision as of 01:15, 17 September 2014

Revision as of 06:22, 18 September 2014

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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]].

==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}}

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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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Moreover, if |''f''(''z'')| = |''z''| for some non-zero ''z'' or |''f′''(0)| = 1, then ''f''(''z'') = ''az'' for some ''a'' in '''C''' with |''a''| = 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===

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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.)

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