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We also need to deal with the fact that the editor who started this article never actually got to the meat of the article: the section "Conformal mapping of Schwarz triangles" is one sentence long! -[[User:Apocheir|Apocheir]] ([[User talk:Apocheir|talk]]) 22:36, 7 October 2021 (UTC)
:In February 2017 the lead originally said:
::<small>In [[mathematics]], the '''Schwarz triangle function''' was introduced by [[H. A. Schwarz]] as the inverse function of the [[conformal mapping]] uniformizing a [[Schwarz triangle]], i.e. a [[hyperbolic triangle|geodesic triangle]] in the [[upper half plane]] with angles which are either 0 or of the form {{pi}} over a positive integer greater than one. 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]]. Through the theory of the [[Schwarzian derivative]], it can be expressed as the quotient of two solutions of a [[hypergeometric differential equation]] with real coefficients and singular points at 0, 1 and ∞. 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 transformation]]s 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 equation]]s with [[regular singular point]]s. In the special case of [[ideal triangle]]s, where all the angles are zero, the tessellation corresponds to the [[Farey series|Farey tessellation]] and the triangle function yields the [[modular lambda function]].</small>
:Later that was modified when another editor inserted content about the hypergeometric ODE tight at the beginning of the lead. As a result crucial conditions on the Schwarz triangles were omitted (that the angles should have the form 0 or {{pi}} over a positive integer) and the geometric connection with tessellations was lost. It is probably a good idea to merge the first version of the lead with the additional content on the hypergeometric function. The densely written paragraph of my original lead can certainly be written in a way that makes it far more approachable to readers.
:Clearly the final section of the article was incomplete. Unfortunately on wikipedia that is often what happens. On the other hand, the theory of the hypergeometric ODE is easy to summarised briefly with exact page references for the books of Caratheodory, Hille and Nehari (already listed in the article). The case of ideal triangles can also be briefly summarised from the 2nd edition of Ahlfors book on Complex Analysis, where the [[modular lambda function]] appears. In this way missing material and page numbers can be added, matching up content on the uniformization problem (conformal mapping of Schwarz triangles) and the hypergeometric ODE. [[User:Mathsci|Mathsci]] ([[User talk:Mathsci|talk]]) 00:43, 8 October 2021 (UTC)
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