Schwarz triangle function: Difference between revisions

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{{short description|Conformal mappings in complex analysis}}

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<!--In [[complex analysis]], the '''Schwarz triangle function''' or '''Schwarz s-function''' is a function that [[conformal mapping|conformally maps]] the [[upper half plane]] to a triangle in the upper half plane having lines or circular arcs for edges. LetThe ''πα'',target ''πβ'',triangle andis ''πγ''not benecessarily thea interior angles at the vertices of the[[Schwarz triangle. If any of ''α]], β'',although andthat ''γ''case are greater than zero, thenis the Schwarzmost trianglemathematically function can be given in terms of [[hypergeometric functions]] as:interesting.

In [[mathematics]], the '''Schwarz triangle tessellation''' was introduced by [[H. A. Schwarz]] as a way of [[tessellation|tessellating]] the hyperbolic [[upper half plane]] by 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. Through 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 hyperbolic refections generates a [[discrete group]], with an orientation-preserving normal subgroup of index two. Each such tessellation yields a [[conformal mapping]] of the upper half plane onto the interior of the geodesic triangle, generalizing the [[Schwarz–Christoffel mapping]], with the upper half plane replacing the complex plane. The corresponding inverse function, first defined by Schwarz, solved the problem of uniformization a hyperbolic triangle and is called the '''Schwarz triangle function'''.

Through the theory of complex [[ordinary differential equation]]s with [[regular singular point]]s and the [[Schwarzian derivative]], the triangle function 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 reflection group 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 [[hypergeometric function]]s.

==Formula==

As the inverse function of such a quotient, the triangle function is an [[automorphic function]] for this discrete group of Möbius transformations. This is a special case of a general scheme of [[Henri Poincaré]] that associates automorphic forms with ordinary differential equations with regular singular points. 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]].

The '''Schwarz triangle function''' or '''Schwarz s-function''' is a function that [[conformal mapping|conformally maps]] the [[upper half plane]] to a triangle in the upper half plane having lines or circular arcs for edges. Let ''πα'', ''πβ'', and ''πγ'' be the interior angles at the vertices of the triangle (in [[radians]]). If any of ''α, β'', and ''γ'' are greater than zero, then the Schwarz triangle function can be given in terms of [[hypergeometric functions]] as:

<!--In [[complex analysis]], the '''Schwarz triangle function''' or '''Schwarz s-function''' is a function that [[conformal mapping|conformally maps]] the [[upper half plane]] to a triangle in the upper half plane having lines or circular arcs for edges. Let ''πα'', ''πβ'', and ''πγ'' be the interior angles at the vertices of the triangle. If any of ''α, β'', and ''γ'' are greater than zero, then the Schwarz triangle function can be given in terms of [[hypergeometric functions]] as:

:<math>s(z) = z^{\alpha} \frac{_2 F_1 \left(a', b'; c'; z\right)}{_2 F_1 \left(a, b; c; z\right)}</math>

where ''a'' = (1−α−β−γ)/2, ''b'' = (1−α+β−γ)/2, ''c'' = 1−α, ''a''′ = ''a'' − ''c'' + 1 = (1+α−β−γ)/2, ''b''′ = ''b'' − ''c'' + 1 = (1+α+β−γ)/2, and ''c''′ = 2 − ''c'' = 1 + α. This mappingformula hascan singularbe pointsderived at ''z'' = 0, 1, and ∞, corresponding tousing the vertices[[Schwarzian of the triangle with angles πα, πγ, and πβ respectivelyderivative]]. At these singular points,

This function can be used to mapmaps the upper half-plane to a [[spherical triangle]] on the [[Riemann sphere]] if α + β + γ > 1, or a [[hyperbolic triangle]] on the [[Poincaré disk]] if α + β + γ < 1. When α + β + γ = 1, then the triangle is a Euclidean triangle with straight edges: ''a''&thinsp;=&thinsp;0, <math>_2 F_1 \left(a, b; c; z\right) = 1</math>, and the formula reduces to that given by the [[Schwarz–Christoffel transformation]]. In the special case of [[ideal triangle]]s, where all the angles are zero, the triangle function yields the [[modular lambda function]].

:<math>\begin{aligned}

s(0) &= 0, \\[5mu]

s(1) &= \frac

{\Gamma(1-a')\Gamma(1-b')\Gamma(c')}

{\Gamma(1-a)\Gamma(1-b)\Gamma(c)},\,\text{and} \\[5mu]

s(\infty) &= \exp\left(i \pi \alpha \right)\frac

{\Gamma(1-a')\Gamma(b)\Gamma(c')}

{\Gamma(1-a)\Gamma(b')\Gamma(c)}.

\end{aligned}</math>

== Value at singular points ==

This formula can be derived using the [[Schwarzian derivative]].

where ''a'' = (1−α−β−γ)/2, ''b'' = (1−α+β−γ)/2, ''c'' = 1−α, ''a''′ = ''a'' − ''c'' + 1 = (1+α−β−γ)/2, ''b''′ = ''b'' − ''c'' + 1 = (1+α+β−γ)/2, and ''c''′ = 2 − ''c'' = 1 + α. This mapping has singular points at ''z'' = 0, 1, and ∞, corresponding to the vertices of the triangle with angles πα, πγ, and πβ respectively. At these singular points,

This function can be used to map the upper half-plane to a [[spherical triangle]] on the [[Riemann sphere]] if α + β + γ > 1, or a [[hyperbolic triangle]] on the [[Poincaré disk]] if α + β + γ < 1. When α + β + γ = 1, then the triangle is a Euclidean triangle with straight edges: ''a''&thinsp;=&thinsp;0, <math>_2 F_1 \left(a, b; c; z\right) = 1</math>, and the formula reduces to that given by the [[Schwarz–Christoffel transformation]]. In the special case of [[ideal triangle]]s, where all the angles are zero, the triangle function yields the [[modular lambda function]].

This function was introduced by [[H. A. Schwarz]] as the inverse function of the [[conformal mapping]] uniformizing a [[Schwarz triangle]]. 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]]. 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.-->

==Conformal mapping of Schwarz triangles==

In this section Schwarz's explicit conformal mapping from the unit disc or the upper half plane to the interior of a Schwarz triangle will be constructed as the ratio of solutions of a hypergeometric ordinary differential equation, following {{harvtxt|Carathéodory|1954|pages=129–194}}, {{harvtxt|Nehari|1975|pages=198–209, 308–332}} and {{harvtxt|Hille|1976|pages=371–401}}. The classical account of {{harvtxt|Ince|1944|pages=389–395}} has an account of the monodromy of second order complex order differential equations with regular singular points. The limiting case of ideal triangles, the [[Farey series]] and the [[modular lambda function]] is explained in {{harvtxt|Ahlfors|1966|pages=254–274}},

{{harvtxt|Chandrasekharan|1985|pages=108–121}}, {{harvtxt|Mumford|Series|Wright|2015}} and {{harvtxt|Hardy|Wright|2008}}.

The '''Schwarz triangle function''' or '''Schwarz s-function''' is a function that [[conformal mapping|conformally maps]] the [[upper half plane]] to a triangle in the upper half plane having lines or circular arcs for edges. Let ''πα'', ''πβ'', and ''πγ'' be the interior angles at the vertices of the triangle. If any of ''α, β'', and ''γ'' are greater than zero, then the Schwarz triangle function can be given in terms of [[hypergeometric functions]] as:

:<math>s(z) = z^{\alpha} \frac{_2 F_1 \left(a', b'; c'; z\right)}{_2 F_1 \left(a, b; c; z\right)}</math>

where ''a'' = (1−α−β−γ)/2, ''b'' = (1−α+β−γ)/2, ''c'' = 1−α, ''a''′ = ''a'' − ''c'' + 1 = (1+α−β−γ)/2, ''b''′ = ''b'' − ''c'' + 1 = (1+α+β−γ)/2, and ''c''′ = 2 − ''c'' = 1 + α. This mapping has singular points at ''z'' = 0, 1, and ∞, corresponding to the vertices of the triangle with angles πα, πγ, and πβ respectively. At these singular points,

:<math>\begin{aligned}

Line 49 ⟶ 27:

\end{aligned}</math>

== Inverse ==

This formula can be derived using the [[Schwarzian derivative]].

As theThe inverse function of such a quotient, the triangle function is an [[automorphic function]] for this discrete group of Möbius transformations. This is a special case of a general scheme of [[Henri Poincaré]] that associates automorphic forms with ordinary differential equations with regular singular points. 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]].

When ''α, β'', and ''γ'' are rational, the triangle is a Schwarz triangle. When ''α, β'', and ''γ'' can each be expressed as the reciprocal of an integer, the triangle is a [[Möbius triangle]], i.e. a non-overlapping Schwarz triangle. When the target triangle is a Möbius triangle, the inverse can be expressed as:

This function can be used to map the upper half-plane to a [[spherical triangle]] on the [[Riemann sphere]] if α + β + γ > 1, or a [[hyperbolic triangle]] on the [[Poincaré disk]] if α + β + γ < 1. When α + β + γ = 1, then the triangle is a Euclidean triangle with straight edges: ''a''&thinsp;=&thinsp;0, <math>_2 F_1 \left(a, b; c; z\right) = 1</math>, and the formula reduces to that given by the [[Schwarz–Christoffel transformation]]. In the special case of [[ideal triangle]]s, where all the angles are zero, the triangle function yields the [[modular lambda function]].

* Spherical: [[rational function]]

* Euclidean: [[elliptical function]]

* Hyperbolic: [[modular function]]

== See also ==

* [[Conformal map projection]]s.<ref name=Lee>{{cite book |last=Lee |first=Laurence |title=Conformal Projections based on Elliptic Functions |year=1976 |publisher=University of Toronto Press |series=Cartographica Monographs |volume=16

|isbn=9780919870161 |url=https://archive.org/details/conformalproject0000leel |url-access=limited }} Chapters also published in [https://www.utpjournals.press/toc/cart/13/1 ''The Canadian Cartographer''. '''13''' (1). 1976.]</ref>

==Notes==

{{notelist}}

==References==