Schwarz triangle function: Difference between revisions

Revision as of 23:15, 2 September 2022

Revision as of 23:49, 2 September 2022

Line 19:

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, aother mappingmaps tocan abe triangleused. withA verticestransformed atform ''i'',of 1,the andSchwarz ''-i'' is given by a transformedtriangle function, with ''α''=''β''=0, ''γ''= 1/2, and <math>z = \frac{1}{1-w^2}</math>., A mappingmaps to aan ideal triangle with vertices at 0''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===