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In complex analysis, the Schwarz triangle function or Schwarz s-function is a function that 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:

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  • Schwarz triangle function
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  • In complex analysis, the Schwarz triangle function or Schwarz s-function is a function that 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:
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  • In complex analysis, the Schwarz triangle function or Schwarz s-function is a function that 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: 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, , and. This formula can be derived using the Schwarzian derivative. 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=0, , and the formula reduces to that given by the Schwarz–Christoffel transformation. In the special case of ideal triangles, 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 transformations 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 equations with regular singular points.
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