Schwarz triangle function: Difference between revisions

Revision as of 22:16, 11 February 2017

Revision as of 14:41, 15 February 2017

Line 1:

In [[mathematics]], the '''Schwarz triangle function''' was introduced by [[H. A. Schwarz]] as the inverse function of the [[conformal mapping]] uniformizing a [[Schwarz triangle]], i.e. a [[hyperbolic triangle|geodesic triangle]] in the [[upper half plane]] with angles which are either 0 or of the form {{pi}} over a positive integer greater than one. Applying successive hyperbolic reflections in its sides, such a triangle generates a [[tessellation]] of the upper half plane (or the unit disk after composition with the [[Cayley transform]]). The conformal mapping of the upper half plane onto the interior of the geodesic triangle generalizes the [[Schwarz–Christoffel transformation]]. Through the theory of the [[Schwarzian derivative]], it can be expressed as the quotient of two solutions of a [[hypergeometric differential equation]] with real coefficients and singular points at 0, 1 and ∞. By the [[Schwarz reflection principle]], the discrete group generated by hyperbolic reflections in the sides of the triangle induces an action on the two dimensional space of solutions. On the orientation-preserving normal subgroup, this two dimensional representation corresponds to the [[monodromy]] of the ordinary differential equation and induces a group of [[Möbius transformation]]s on quotients of solutions. Since the triangle function is the inverse function of such a quotient, it is therefore an [[automorphic function]] for this discrete group of Möbius transformations. This is a special case of a general method of [[Henri Poincaré]] that associates automorphic forms with [[ordinary differential equation]]s with [[regular singular point]]s. In the special case of [[ideal triangle]]s, where all the angles are zero, the tessellation corresponds to the [[Farey series|Farey tessellation]] and the triangle function yields the [[modular lambda function]].

==Hyperboloid and Klein models==

==Convex polygons==

==Tessellation by Schwarz triangles==

In this section tessellations of the hyperbolic upper half plane by Schwarz triangles will be discussed using elementary methods. For triangles with ideal vertices, i.e. where all three angles are strictly positive, the elementary approach of {{harvtxt|Caratheodory|1954}} will be followed. For triangles with one or two ideal vertices, i.e. one or two angles equal to zero, elementary arguments of {{harvtxt|Evans|1973}}, simplifying the approach of {{harvtxt|Hecke|1935}}, will be used: in the case of a Schwarz triangle with one angle zero and another a right angle, the orientation-preserving subgroup of the reflection group of the triangle is a [[Hecke group]]. For an ideal triangle in which all angles are zero, the existence of the tessellation will be established by relating it to the [[Farey series]] described in {{harvtxt|Hardy|Wright|1979}} and {{harvtxt|Series|2015}}. In this case the tessellation can be considered as that associated with three touching circles on the [[Riemann sphere]], a limiting case of configurations associated with three disjoint non-nested circles and their reflection groups, the so-called "[[Schottky group]]s", described in detail in {{harvtxt|Mumford|Series|Wright|2015}}.

===Triangles with no ideal vertices===

===Triangles with one or two ideal vertices===

===Ideal triangles===

==References==

*{{citation|last=Beardon|first=Alan F.|title=The geometry of discrete groups|series=Graduate Texts in Mathematics|volume= 91|publisher =

Line 9 ⟶ 15:

*{{citation|last=Davis|first= Michael W.|title=The geometry and topology of Coxeter groups|series=London Mathematical Society Monographs|volume=32|publisher= Princeton University Press|year=2008|isbn= 978-0-691-13138-2}}

*{{citation|last=de Rham|first=G.|authorlink=Georges de Rham|title=Sur les polygones générateurs de groupes fuchsiens|journal= Enseignement Math.|volume= 17 |year=1971|pages= 49–61}}

*{{citation|last=Evans|first= Ronald|title=A fundamental region for Hecke's modular group|journal=J. Number Theory|volume= 5|year=1973|pages=108–115}}

*{{citation|title=Automorphic Functions|first=Lester R. |last=Ford|authorlink=Lester R. Ford|publisher=[[American Mathematical Society]]|year= 1951|isbn=0821837419}}, reprint of 1929 edition

*{{citation|last=Hardy|first=G. H.|authorlink=G. H. Hardy|last2=Wright|first2= E. M.|authorlink2=E. M. Wright|title=An introduction to the theory of numbers|edition=Fifth|[[Oxford University Press]]| year=1979|isbn= 0-19-853170-2}}

*{{citation|title=Über die Bestimmung Dirichletscher Reihen durch ihre Funktionalgleichung|language=de|last=Hecke|first= E.|authorlink=Erich Hecke|journal=Mathematische Annalen | volume=112|year=1935|pages=664–699}}

*{{citation|last=Helgason|first=Sigurdur|authorlink=Sigurdur Helgason|title= Groups and geometric analysis. Integral geometry, invariant differential operators, and spherical functions|series= Mathematical Surveys and Monographs|volume=83|publisher=[[American Mathematical Society]]|year= 2000|isbn=0-8218-2673-5}}

*{{citation|last=Hille|first= Einar|authorlink=Einar Hille|title=Ordinary differential equations in the complex ___domain|publisher=Wiley-Interscience|year= 1976}}

Line 18 ⟶ 27:

|last1=Mumford|first1=David|authorlink1=David Mumford|last2=Series|first2= Caroline|authorlink2=Caroline Series|last3=Wright|first3=David}}

*{{citation|last=Nehari|first=Zeev|authorlink=Zeev Nehari|title= Conformal mapping|publisher=Dover Publications|year= 1975}}

*{{citation|last=Series|first=Caroline|authorlink=Caroline Series|title=The modular surface and continued fractions|journal=J. London Math. Soc.| volume= 31 |year=1985|pages=69–80}}

*{{citation|last=Series|first=Caroline|authorlink=Caroline Series|title=Continued fractions and hyperbolic geometry, Loughborough LMS Summer School|year=2015|url=http://homepages.warwick.ac.uk/~masbb/HypGeomandCntdFractions-2.pdf|accessdate=15 February 2017}}

*{{citation|last=Siegel| first=C. L.|authorlink=Carl Ludwig Siegel|title=Topics in complex function theory, Vol. II. Automorphic functions and abelian integrals|translator1=A. Shenitzer|translator2= M. Tretkoff|publisher=Wiley-Interscience|year=1971|isbn=0-471-60843-2}}

*{{citation|last=Thurston|first=William P.|authorlink=William Thurston|title=Three-dimensional geometry and topology. Vol. 1.|editor= Silvio Levy|series= Princeton Mathematical Series|volume=35|publisher=[[Princeton University Press]] |year=1997|isbn= 0-691-08304-5}}