Schwarz triangle function: Difference between revisions

Revision as of 01:22, 4 September 2022

Revision as of 18:37, 5 September 2022

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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'', but that is not an option for an [[ideal triangle]] having all angles zero.

AInstead, Schwarza triangle function with transformed input mapsmapping to an ideal triangle with vertices at ''i''0, 1, and ''-i''∞ is given by in terms of the [[complete elliptic integral of the first kind]]:

: <math>s(0, 0, \frac{1}{2}; \frac{1}{1-z^2})</math>

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]]:

:<math>i\frac{K(1-z)}{K(z)}</math>.

TheThis inverse of this functionexpression is athe namedinverse function,of the [[modular lambda function]].{{sfn|Nehari|1975|pp=316-318}}

== Extensions ==