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{{short description|Conformal mappings in complex analysis}}
[[File:Schwarz triangle function.svg|thumb|The upper half-plane, and the image of the upper half-plane transformed by the Schwarz triangle function with various parameters.]]
{{Complex analysis sidebar}}
In [[complex analysis]], the '''Schwarz triangle function''' or '''Schwarz s-function''' is a function that [[conformal mapping|conformally maps]] the [[upper half plane]] to a triangle in the upper half plane having lines or circular arcs for edges. The target triangle is not necessarily a [[Schwarz triangle]], although that is the most mathematically interesting case. When that triangle is a non-overlapping Schwarz triangle, i.e. a [[Möbius triangle]], the inverse of the Schwarz triangle function is a [[single-valued]] [[automorphic function]] for that triangle's [[triangle group]]. More specifically, it is a [[modular function]].
==Formula==
Let ''πα'', ''πβ'', and ''πγ'' be the interior angles at the vertices of the triangle in [[radians]]. Each of ''α'', ''β'', and ''γ'' may take values between 0 and 1 inclusive. Following Nehari,{{sfn|Nehari|1975|page=309}} these angles are in clockwise order, with the vertex having angle ''πα'' at the origin and the vertex having angle ''πγ'' lying on the real line. The Schwarz triangle function can be given in terms of [[hypergeometric functions]] as:
:<math>s(\alpha, \beta, \gamma; z) = z^{\alpha} \frac{_2 F_1 \left(a', b'; c'; z\right)}{_2 F_1 \left(a, b; c; z\right)}</math>
where
:''a'' = (1−α−β−γ)/2,
:''b'' = (1−α+β−γ)/2,
:''c'' = 1−α,
:''a''′ = ''a'' − ''c'' + 1 = (1+α−β−γ)/2,
:''b''′ = ''b'' − ''c'' + 1 = (1+α+β−γ)/2, and
:''c''′ = 2 − ''c'' = 1 + α.
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'' = 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]].
===Derivation===
Through the theory of complex [[ordinary differential equation]]s with [[regular singular point]]s and the [[Schwarzian derivative]], the triangle function 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 reflection group 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 [[hypergeometric function]]s.{{sfn|Nehari|1975|pp=198-208}}
== 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>\begin{align}
s(0) &= 0, \\[4mu]
s(1) &= \frac
{\Gamma(1-a')\Gamma(1-b')\Gamma(c')}
{\Gamma(1-a)\Gamma(1-b)\Gamma(c)}, \\[8mu]
s(\infty) &= \exp\left(i \pi \alpha \right)\frac
{\Gamma(1-a')\Gamma(b)\Gamma(c')}
{\Gamma(1-a)\Gamma(b')\Gamma(c)},
\end{align}</math>
where <math display=inline>\Gamma(x)</math> is the [[gamma function]].
Near each singular point, the function may be approximated as
:<math>\begin{align}
s_0(z) &= z^\alpha (1+O(z)), \\[6mu]
s_1(z) &= (1-z)^\gamma (1+O(1-z)), \\[6mu]
s_\infty(z) &= z^\beta (1+O(1/z)),
\end{align}</math>
where <math>O(x)</math> is [[big O notation]].
== Inverse ==
When ''α, β'', and ''γ'' are rational, the triangle is a Schwarz triangle. When each of ''α, β'', and ''γ'' are either the reciprocal of an integer or zero, the triangle is a [[Möbius triangle]], i.e. a non-overlapping Schwarz triangle. For a Möbius triangle, the inverse is a [[modular function]].
In the spherical case, that modular function is a [[rational function]]. For Euclidean triangles, the inverse can be expressed using [[elliptical function]]s.<ref name=Lee />
== Ideal triangles ==
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.
Instead, 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>.
This expression is the inverse of the [[modular lambda function]].{{sfn|Nehari|1975|pp=316-318}}
== Extensions ==
The [[Schwarz–Christoffel transformation]] gives the mapping from the upper half-plane to any Euclidean polygon.
The methodology used to derive the Schwarz triangle function earlier can be applied more generally to arc-edged polygons. However, for an ''n''-sided polygon, the solution has ''n'' − 3 additional parameters, which are difficult to determine in practice.{{sfn|Nehari|1975|p=202}} See {{slink|Schwarzian derivative#Conformal mapping of circular arc polygons}} for more details.
== Applications ==
[[L. P. Lee]] used Schwarz triangle functions to derive [[conformal map projection]]s onto [[polyhedral map projection|polyhedral]] surfaces.<ref name=Lee>{{cite book | last = Lee | first = L. P. | author-link = Laurence Patrick Lee | year = 1976 | title = Conformal Projections Based on Elliptic Functions | ___location = Toronto | publisher = B. V. Gutsell, York University | series = Cartographica Monographs | volume = 16 | url = https://archive.org/details/conformalproject0000leel | url-access = limited | isbn = 0-919870-16-3}} Supplement No. 1 to [https://www.utpjournals.press/toc/cart/13/1 ''The Canadian Cartographer'' '''13'''].</ref>
==References==
{{reflist|30em}}
*{{citation|last=Caratheodory|first=C.|authorlink=Constantin Caratheodory|title=Theory of functions of a complex variable. Vol. 2.|translator=F. Steinhardt. |publisher=Chelsea Publishing Company|year= 1954}}▼
==Sources==
{{refbegin}}
* {{
▲* {{
* {{Cite book |last=Hille |first=Einar |author-link=Einar Hille |title=Ordinary differential equations in the complex ___domain |date=1997 |publisher=Dover Publications |isbn=0-486-69620-0 |___location=Mineola, N.Y. |oclc=36225146}}
* {{
* {{cite book |last1=Sansone |first1=Giovanni |author-link=Giovanni Sansone |last2=Gerretsen |first2=Johan |title=Lectures on the theory of functions of a complex variable. II: Geometric theory |publisher=Wolters-Noordhoff |year=1969 |oclc=245996162}}
{{refend}}
[[Category:Complex analysis]]
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[[Category:Modular forms]]
[[Category:Spherical geometry]]
[[Category:Automorphic forms]]
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