Schwarz triangle function: Difference between revisions

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[[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 case 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,

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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''&thinsp;=&thinsp;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]].

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''. For an [[ideal triangle]] having all angles zero, other maps can be used. A transformed form of the Schwarz triangle function, with ''α''=''β''=0, ''γ''= 1/2, and <math>z = \frac{1}{1-w^2}</math>, maps to an ideal triangle with vertices at ''i'', 1, and ''-i''. Alternately, 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]]:{{sfn|Nehari|1975|pp=316-318}}

:<math>i\frac{K(1-z)}{K(z)}</math>,

which is the inverse of the [[modular lambda function]].

===Derivation===

Line 38 ⟶ 34:

\end{align}</math>

where <math display=inline>\Gamma(x)</math> is the [[Gammagamma function]].

Near each singular point, the function may be approximated as

Line 48 ⟶ 44:

\end{align}</math>

where <math>O(x)</math> is [[Bigbig 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. When the target triangle isFor a Möbius triangle, the inverse canis a be[[modular expressedfunction]]. as:

The inverse function is an [[automorphic function]] for this discrete group of Möbius transformations. This is a special case of a general scheme of [[Henri Poincaré]] that associates automorphic forms with ordinary differential equations with regular singular points.

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, 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. When the target triangle is a Möbius triangle, the inverse can be expressed as:

* Spherical: [[rational function]]

== Ideal triangles ==

* Euclidean: [[elliptical function]]

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.

* Hyperbolic: [[modular function]]

** Ideal triangle: [[modular lambda function]]

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>,.

whichThis 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'' − 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 [[polyhedralconformal map projection]]s that areonto [[conformalpolyhedral map projection|conformalpolyhedral]] surfaces.<ref name=Lee>{{cite book | last = Lee | first = L. P. | author-link = Laurence Patrick Lee | year = 1976 | title = Conformal Projections basedBased on Elliptic Functions |year ___location =1976 Toronto | publisher =University ofB. TorontoV. PressGutsell, 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>

|isbn=9780919870161 |url=https://archive.org/details/conformalproject0000leel |url-access=limited }} Chapters also published in [https://www.utpjournals.press/toc/cart/13/1 ''The Canadian Cartographer''. '''13''' (1). 1976.]</ref>

==References==

{{reflist|20em30em}}

==Sources==

{{refbegin|colwidth=30em}}

* {{citationCite book |last=Ahlfors |first=Lars V. |author-link=Lars Ahlfors |title=Complex Analysisanalysis: an introduction to the theory of analytic functions of one complex variable |date=1979 |publisher=McGraw -Hill |yearisbn=19660-07-000657-1 |edition=2nd3 |___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}}

* {{citation |last=Ahlfors |first=Lars V. |author-link=Lars Ahlfors |title=Complex Analysis |publisher=McGraw Hill |year=1966 |edition=2nd }}

* {{citationCite book |last=CarathéodoryHille |first=ConstantinEinar |author-link=ConstantinEinar CarathéodoryHille |title=TheoryOrdinary ofdifferential functionsequations ofin athe complex variable___domain |volumedate=21997 |translatorpublisher=F.Dover SteinhardtPublications |publisherisbn=Chelsea0-486-69620-0 |___location=Mineola, N.Y. |yearoclc=195436225146}}

* {{citationCite book |last=ChandrasekharanNehari |first=K.Zeev |author-link=K.Zeev S. ChandrasekharanNehari |title=EllipticConformal functionsmapping |seriesdate=Grundlehren der Mathematischen Wissenschaften |volume=2811975 |publisher=SpringerDover |year=1985Publications |isbn=30-540486-1529561137-4X |___location=New York |oclc=1504503}}

* {{citationcite book |last1=HardySansone |first1=G. H.Giovanni |author-link=G.Giovanni H. HardySansone |last2=WrightGerretsen |first2=E. M. |author-link2=E. M. WrightJohan |title=AnLectures introduction toon the theory of numbersfunctions |edition=6thof a complex variable. II: Geometric theory |publisher=Oxford University PressWolters-Noordhoff |year=20081969 |isbnoclc=978-0-19-921986-5 245996162}}

* {{citation |last=Hille |first=Einar |author-link=Einar Hille |title=Ordinary differential equations in the complex ___domain |publisher=Wiley-Interscience |year=1976 }}

* {{citation |last=Ince |first=E. L. |author-link=E. L. Ince |title=Ordinary Differential Equations |publisher=Dover |year=1944 }}

* {{citation |last1=Mumford |first1=David |author-link1=David Mumford |last2=Series |first2=Caroline |author-link2=Caroline Series |last3=Wright |first3=David |date=2015 |title=Indra's pearls. The vision of Felix Klein |publisher=Cambridge University Press |isbn=978-1-107-56474-9 }}

* {{citation |last=Nehari |first=Zeev |author-link=Zeev Nehari |title=Conformal mapping |publisher=Dover |year=1975 }}

{{refend}}

== Further reading ==

* {{citation |title=Automorphic Functions |first=Lester R. |last=Ford |author-link=Lester R. Ford |publisher=[[American Mathematical Society]] |year=1951 |isbn=0821837419 |orig-date=1929}}

* {{citation |last=Lehner |first=Joseph |author-link=Joseph Lehner |title=Discontinuous groups and automorphic functions |series=Mathematical Surveys |volume=8 |publisher=American Mathematical Society |year=1964 }}. (Note that Lehner has pointed out that his proof of Poincaré's polygon theorem is incomplete. He has subsequently recommended de Rham's 1971 exposition.)

* {{citation |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 }}

* {{citation |last=Series |first=Caroline |author-link=Caroline Series |title=The modular surface and continued fractions |journal=Journal of the London Mathematical Society |volume=31 |year=1985 |pages=69–80 |doi=10.1112/jlms/s2-31.1.69 }}

* {{citation |last=Thurston |first=William P. |author-link=William Thurston |title=Three-dimensional geometry and topology. Vol. 1. |editor=Silvio Levy |series=Princeton Mathematical Series |volume=35 |publisher=Princeton University Press |year=1997 |isbn=0-691-08304-5 }}

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