Content deleted Content added
→Value at singular points: split up the math tag, aligning doesn't help much here |
→Formula: listify the equations for a b c etc. |
||
Line 9:
:<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
:''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 formula can be derived using the [[Schwarzian derivative]]. 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]].
| |||