Schwarz triangle function

<p>In <a href="/facts/Complex_analysis/hgCoQlH0">complex analysis</a>, the Schwarz triangle function or Schwarz s-function is a function that <a href="/facts/Conformal_mapping/cEpQnMad">conformally maps</a> the <a href="/facts/Upper_half_plane/7fPqrowr">upper half plane</a> to a triangle in the upper half plane having lines or circular arcs for edges. The target triangle is not necessarily a <a href="/facts/Schwarz_triangle/bIz52h1W">Schwarz triangle</a>, although that is the most mathematically interesting case. When that triangle is a non-overlapping Schwarz triangle, i.e. a <a href="/facts/M%C3%B6bius_triangle/bIz52h1W">Möbius triangle</a>, the inverse of the Schwarz triangle function is a <a href="/facts/Single-valued/WH5ua6BL">single-valued</a> <a href="/facts/Automorphic_function/5L59Quh8">automorphic function</a> for that triangle's <a href="/facts/Triangle_group/LrEs3xVw">triangle group</a>. More specifically, it is a <a href="/facts/Modular_function/PcnbYR1f">modular function</a>.
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Schwarz triangle function open-in-new

Schwarz triangle function