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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
==Formula==
Let ''πα'', ''πβ'', and ''πγ'' be the interior angles at the vertices of the triangle
:<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,
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:''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)}
{\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 />
==
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|
==Sources==
{{refbegin
* {{Cite book |last=Ahlfors |first=Lars V. |author-link=Lars Ahlfors |title=Complex analysis: an introduction to the theory of analytic functions of one complex variable |date=1979 |publisher=McGraw-Hill |isbn=0-07-000657-1 |edition=3 |___location=New York |oclc=4036464}}
* {{cite book |last=Carathéodory |first=Constantin |author-link=Constantin Carathéodory |title=Theory of functions of a complex variable |volume=2 |translator=F. Steinhardt |publisher=Chelsea |year=1954|url=https://books.google.com/books?id=UTTvAAAAMAAJ|oclc=926250115}}
* {{
* {{
* {{
{{refend}}
[[Category:Complex analysis]]
[[Category:Hyperbolic geometry]]
[[Category:Conformal mappings]]
[[Category:Modular forms]]
[[Category:Spherical geometry]]
[[Category:Automorphic forms]]
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