Schwarz triangle function: Difference between revisions

Revision as of 15:37, 29 August 2022

Revision as of 17:29, 29 August 2022

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== Singular points ==

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>, \begin{align}

:<math>s(10) &= 0, \\frac[4mu]

s(1) &= \frac

{\Gamma(1-a')\Gamma(1-b')\Gamma(c')}

{\Gamma(1-a)\Gamma(1-b)\Gamma(c)}</math>, and \\[8mu]

:<math>s(\infty) &= \exp\left(i \pi \alpha \right)\frac

{\Gamma(1-a')\Gamma(b)\Gamma(c')}

{\Gamma(1-a)\Gamma(b')\Gamma(c)}</math>,

\end{align}</math>

where Γ<math display=inline>\Gamma(x)</math> 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>,begin{align}

:<math>s_1s_0(z) &=(1- z)^\gammaalpha (1+O(1-z))</math>, and\\[6mu]

:<math>s_\inftys_1(z) &= (1-z)^\betagamma (1+O(\tfrac{1}{-z}))</math>., \\[6mu]

s_\infty(z) &= z^\beta (1+O(1/z)),

\end{align}</math>

where <math>O(x)</math> is [[Big O notation]].

== Inverse ==