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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.-->
==Hyperboloid and Beltrami-Klein models==
[[File:Relation5models.png|thumb|upright=0.85|Geometric relations between Poincaré disk model, hyperboloid model and Beltrami-Klein model]]
In this section two different models are given for hyperbolic geometry on the unit disk or equivalently the upper half plane.<ref>See:
*{{harvnb|Busemann|1955}}
*{{harvnb|Magnus|1974}}
*{{harvnb|Helgason|2000}}
*{{harvnb|Wolf|2011}}</ref>
The group ''G'' = SU(1,1) is formed of matrices
:<math> g = \begin{pmatrix} \alpha & \beta \\ \overline{\beta} & \overline{\alpha}\end{pmatrix}
</math>
with
:<math> |\alpha|^2 -|\beta|^2=1.</math>
It is a subgroup of ''G''<sub>''c''</sub> = SL(2,'''C'''), the group of complex 2 × 2 matrices with determinant 1.
The group ''G''<sub>''c''</sub> acts by Möbius transformations on the extended complex plane. The subgroup ''G'' acts as automorphisms of the unit disk ''D'' and the subgroup ''G''<sub>1</sub> = SL(2,'''R''') acts as automorphisms of the [[upper half plane]]. If
:<math>C=\begin{pmatrix}1 & i \\ i & 1\end{pmatrix}</math>
then
:<math>G=CG_1C^{-1},</math>
since the Möbius transformation corresponding ''M'' is the [[Cayley transform]] carrying the upper half plane onto the unit disk and the real line onto the unit circle.
The Lie algebra <math>\mathfrak{g}</math> of SU(1,1) consists of matrices
:<math> X= \begin{pmatrix} ix & w \\ \overline{w} & -ix \end{pmatrix}</math>
with ''x'' real. Note that ''X''<sup>2</sup> = (|''w''|<sup>2</sup> – ''x''<sup>2</sup>) ''I'' and
:<math> \det X = x^2 -|w|^2 =-\tfrac12 \operatorname{Tr} X^2.</math>
The hyperboloid <math>\mathfrak{H}</math> in <math>\mathfrak{g}</math> is defined by two conditions. The first is that det ''X'' = 1 or equivalently Tr ''X''<sup>2</sup> = –2.<ref>Note that the [[Killing form]] ''B''(''X,''Y'') = Tr<math>{}_{\mathfrak g}</math> ad ''X'' ad ''Y'' = 4 Tr ''XY''.{{harvnb|Helgason|1978}}</ref> By definition this condition is preserved under [[Conjugacy class|conjugation]] by ''G''. Since ''G'' is connected it leaves the two components with ''x'' > 0 and ''x'' < 0 invariant. The second condition is that ''x'' > 0. For brevity, write ''X'' = (''x'',''w'').
The group ''G'' acts transitively on ''D'' and <math>\mathfrak{H}</math> and the points 0 and (1,0) have [[Stabilizer subgroup|stabiliser]] ''K'' consisting of matrices
:<math>\begin{pmatrix}\zeta & 0 \\ 0 & \overline{\zeta}\end{pmatrix}</math>
with |ζ| = 1. Polar decomposition on ''D'' implies the Cartan decomposition ''G'' = ''KAK'' where ''A'' is the group of matrices
:<math>a_t = \begin{pmatrix}\cosh t & \sinh t \\ \sinh t & \cosh t\end{pmatrix}.</math>
Both spaces can therefore be identified with the homogeneous space ''G''/''K'' and there is a ''G''-equivariant map ''f'' of <math>\mathfrak{H}</math> onto ''D'' sending (1,0) to 0. To work out the formula for this map and its inverse it suffices to compute ''g''(1,0) and
''g''(0) where ''g'' is as above. Thus ''g''(0) = β/{{overline|α}} and
:<math>g(1,0)=(|\alpha|^2 +|\beta|^2,-2i\alpha\beta),</math>
so that
:<math>\frac\beta\overline{\alpha} = \frac{\alpha\beta} {|\alpha|{}^2} = \frac{2\alpha\beta}{|\alpha|{}^2 + |\beta|{}^2 + 1},</math>
recovering the formula
:<math>f(x,w) = \frac{iw}{x + 1}.</math>
Conversely if ''z'' = ''iw''/(''x'' + 1), then |''z''|<sup>2</sup> = (''x'' – 1)/(''x'' + 1), giving the inverse formula
:<math> f^{-1}(z) = (x,w) = \left( \frac{1 + |z|{}^2}{1 - |z|{}^2}, \frac{-2iz}{1 - |z|{}^2} \right).</math>
This correspondence extends to one between geometric properties of ''D'' and <math>\mathfrak{H}</math>. Without entering into the correspondence of ''G''-invariant [[Riemannian metric]]s,{{efn|1=The Poincaré metric on the disk corresponds to the restriction of the ''G''-invariant pseudo-Riemannian metric ''dx''<sup>2</sup> – ''dw''<sup>2</sup> to the hyperboloid.}} each geodesic circle in ''D'' corresponds to the intersection of 2-planes through the origin, given by equations Tr ''XY'' = 0, with <math>\mathfrak{H}</math>. Indeed, this is obvious for rays arg ''z'' = θ through the origin in ''D''—which correspond to the 2-planes arg ''w'' = θ—and follows in general by ''G''-equivariance.
==Conformal mapping of Schwarz triangles==
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