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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'' = 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]].
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This mapping has [[regular 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,{{sfn|Nehari|1975|pages=315−316}}
:<math>s(0) = 0</math>,
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where Γ(x) is the [[Gamma function]].
Near each singular point, the function may be approximated as so, using [[Big O notation]].
:<math>s_0(z)=z^\alpha (1+O(z))</math>,
:<math>s_1(z)=(1-z)^\beta (1+O(1-z))</math>, and
:<math>s_\infty(z)=z^\gamma (1+O(\tfrac{1}{z}))</math>.
== Inverse ==
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