Schwarz triangle function: Difference between revisions

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Let ''πα'', ''πβ'', and ''πγ'' be the interior angles at the vertices of the triangle in [[radians]]. Each of ''α'', ''β'', and ''γ'' may take values between 0 and 1 inclusive. Following Nehari,{{sfn|Nehari|1975|page=309}} these angles are in clockwise order, with the vertex having angle ''πα'' at the origin and the vertex having angle ''πγ'' lying on the real line. The Schwarz triangle function can be given in terms of [[hypergeometric functions]] as:

:<math>s(\alpha, \beta, \gamma; 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,

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This function maps the upper half-plane to a [[spherical triangle]] if α + β + γ > 1, or a [[hyperbolic triangle]] 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]].

When ''α'' = 0 the triangle is degenerate, lying entirely on the real line. If either of ''β'' or ''γ'' are non-zero, the angles can be permuted so that the positive value is ''α''. For an [[ideal triangle]] having all angles zero, other maps can be used. A transformed form of the Schwarz triangle function, with ''α''=''β''=0, ''γ''= 1/2, and <math>z = \frac{1}{1-w^2}</math>, maps to an ideal triangle with vertices at ''i'', 1, and ''-i''. Alternately, a mapping to an ideal triangle with vertices at 0, 1, and ∞ is given by in terms of the [[complete elliptic integral of the first kind]]:{{sfn|Nehari|1975|pp=316-318}}

:<math>i\frac{K(1-z)}{K(z)}</math>,

which is the inverse of the [[modular lambda function]].

===Derivation===

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== Inverse ==

When ''α, β'', and ''γ'' are rational, the triangle is a Schwarz triangle. When each of ''α, β'', and ''γ'' are either the reciprocal of an integer or zero, the triangle is a [[Möbius triangle]], i.e. a non-overlapping Schwarz triangle. For a Möbius triangle, the inverse is a [[single-valued]] function, which canare be[[automorphic expressedfunction]]s using:for the [[triangle group]] of Möbius transformations for the given Möbius triangle.

Depending on the geometry of the triangle, the inverse function can be expressed using:

* Spherical: [[rational function]]s

* Euclidean: [[elliptical function]]s

* Hyperbolic: [[modular function]]s

These inverse functions are [[automorphic function]]s for the [[triangle group]] of Möbius transformations for the given Möbius triangle.

== Ideal triangles ==

As mentioned earlier, the Schwarz triangle function is not directly useful for ideal triangles, but the [[modular lambda function]] can be used to express maps from ideal triangles to the upper half-plane.

When ''α'' = 0 the triangle is degenerate, lying entirely on the real line. If either of ''β'' or ''γ'' are non-zero, the angles can be permuted so that the positive value is ''α'', but that is not an option for an [[ideal triangle]] having all angles zero.

A Schwarz triangle function with transformed input maps to an ideal triangle with vertices at ''i'', 1, and ''-i'':

: <math>s(0, 0, \frac{1}{2}; \frac{1}{1-z^2})</math>

Alternately, a mapping to an ideal triangle with vertices at 0, 1, and ∞ is given by in terms of the [[complete elliptic integral of the first kind]]:

:<math>i\frac{K(1-z)}{K(z)}</math>,.

The inverse of this function is a named function, the [[modular lambda function]].{{sfn|Nehari|1975|pp=316-318}}

== Extensions ==

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* {{Cite book |last=Nehari |first=Zeev |author-link=Zeev Nehari |title=Conformal mapping |date=1975 |publisher=Dover Publications |isbn=0-486-61137-X |___location=New York |oclc=1504503}}

* {{cite book |last1=Sansone |first1=Giovanni |author-link=Giovanni Sansone |last2=Gerretsen |first2=Johan |title=Lectures on the theory of functions of a complex variable. II: Geometric theory |publisher=Wolters-Noordhoff |year=1969 |oclc=245996162}}

{{refend}}