Schwarz triangle function: Difference between revisions

Revision as of 22:19, 24 April 2022

Revision as of 01:06, 6 July 2022

Line 487:

== See also ==

* [[Conformal map projection]]s.<ref name=Lee>{{cite book |last=Lee |first=Laurence |title=Conformal Projections based on Elliptic Functions |year=1976 |publisher=University of Toronto Press |series=Cartographica Monographs |volume=16

|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>

==Notes==

Line 504:

* {{citation |last1=Berndt |first1=Bruce C. |author-link=Bruce C. Berndt |last2=Knopp |first2=Marvin I. |author-link2=Marvin Knopp |title=Hecke's theory of modular forms and Dirichlet series |series=Monographs in Number Theory |volume=5 |publisher=World Scientific |year=2008 |isbn=978-981-270-635-5 }}

* {{cite book | mr =2133266 |last1= Björner| first1= Anders| author1-link= Anders Björner |

last2= Brenti| first2= Francesco | title=Combinatorics of Coxeter groups| series= Graduate Texts in Mathematics| volume= 231| publisher=[[Springer-Verlag]]| year= 2005 | isbn = 978-3540-442387}}

* {{cite book| mr =1482800 | last = Borel | first = Armand | author-link = Armand Borel | title = Automorphic forms on SL₂(R) | series = Cambridge Tracts in Mathematics | volume = 130

Combinatorics of Coxeter groups| series= Graduate Texts in Mathematics| volume= 231| publisher=

[[Springer-Verlag]]| year= 2005 | isbn = 978-3540-442387}}

* {{cite book| mr =1482800 | last = Borel | first = Armand | author-link = Armand Borel | title =

Automorphic forms on SL₂(R) | series = Cambridge Tracts in Mathematics | volume = 130

| publisher = [[Cambridge University Press]] | year = 1997 | isbn = 0-521-58049-8 | url = https://www.cambridge.org/core/books/automorphic-forms-on-sl2-r/20CDEAD276B61040B2B9013A21ACC145

|chapter= I. Basic material on SL₂(R), discrete subgroups, and the upper half-plane}}

* {{cite book| mr =0240238|last=Bourbaki|first= Nicolas|author-link= Nicolas Bourbaki |series=Éléments de mathématique |title=Groupes et algèbres de Lie|chapter= Chapitre IV : Groupes de Coxeter et systèmes de Tits • Chapitre V : Groupes engendrés par des réflexions|publisher= Hermann|___location= Paris|year = 1968 | pages =1–56, 57–141 | lang = fr}} Reprinted by Masson in 1981 as {{isbn|2-225-76076-4}}.

* {{cite book |mr =1744486 | last1=Bridson| first1= Martin R.|author1-link= Martin Bridson|last2= Haefliger| first2 = André | author2-link = André Haefliger | title = Metric spaces of non-positive curvature | series =Grundlehren der mathematischen Wissenschaften | volume = 319| publisher = [[Springer-Verlag]] | year = 1999

Éléments de mathématique |title=Groupes et algèbres de Lie|chapter= Chapitre IV : Groupes de Coxeter et systèmes de Tits • Chapitre V : Groupes engendrés par des réflexions|publisher= Hermann|___location= Paris|year = 1968 | pages =1–56, 57–141 | lang = fr}} Reprinted by Masson in 1981 as {{isbn|2-225-76076-4}}.

| isbn = 3-540-64324-9 | url = https://www.math.bgu.ac.il/~barakw/rigidity/bh.pdf |chapter =I. Basic material on SL₂(R), discrete subgroups, and the upper half-plane}}

* {{cite book |mr =1744486 | last1=Bridson| first1= Martin R.|author1-link= Martin Bridson|last2= Haefliger| first2 = André | author2-link = André Haefliger | title = Metric spaces of non-positive curvature | series =

Grundlehren* der{{cite mathematischenbook|mr = Wissenschaften0969123 | volumelast = 319Brown | first = Kenneth S. | author-link = Kenneth S. Brown| title=Buildings| publisher = [[Springer-Verlag]] | year = 1999 1989 | isbn = 0-387-96876-8 }}

| isbn = 3-540-64324-9 | url = https://www.math.bgu.ac.il/~barakw/rigidity/bh.pdf |chapter =

I. Basic material on SL₂(R), discrete subgroups, and the upper half-plane}}

* {{cite book|mr = 0969123 | last = Brown | first = Kenneth S. | author-link = Kenneth S. Brown| title=

Buildings| publisher = [[Springer-Verlag]] | year = 1989 | isbn = 0-387-96876-8 }}

* {{citation |last=Busemann |first=Herbert |author-link=Herbert Busemann |title=The geometry of geodesics |publisher=Academic Press |year=1955 }}

* {{citation |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}}

* {{citation |last=Chandrasekharan |first=K. |author-link=K. S. Chandrasekharan |title=Elliptic functions |series=Grundlehren der Mathematischen Wissenschaften |volume=281 |publisher=Springer |year=1985 |isbn=3-540-15295-4 }}

* {{citation |last=Davis |first=Michael W. |title=The geometry and topology of Coxeter groups |series=London Mathematical Society Monographs |volume=32 |publisher=[[Princeton University Press]] |year=2008 |isbn=978-0-691-13138-2| chapter=Appendix D. The Geometric Representation| pages = 439–447 }}

* {{cite book|mr=1170367| last=de la Harpe|first= Pierre | chapter=An invitation to Coxeter groups|title= Group theory from a geometrical viewpoint (Trieste, 1990)|pages= 193–253|publisher= [[World Scientific]] |year= 1991}}

* {{cite journal| mr =0647210 |last=Deodhar | first= Vinay V. | author-link= Vinay Deodhar| title = On the root system of a Coxeter group |journal = [[Comm. Algebra]]| volume = 10 | year =1982 |issue=6 | pages = 611–630 |

An invitation to Coxeter groups|title= Group theory from a geometrical viewpoint (Trieste, 1990)|pages= 193–253|publisher= [[World Scientific]] |year= 1991}}

* {{cite journal| mr =0647210 |last=Deodhar | first= Vinay V. | author-link= Vinay Deodhar| title = On the root system of a Coxeter group |journal = [[Comm. Algebra]]| volume = 10 | year =1982 | pages = 611–630 |

doi=10.1080/00927878208822738}}

* {{cite journal| url = https://www.e-periodica.ch/cntmng?var=true&pid=ens-001:1986:32::54| last=Deodhar | first= Vinay V. | author-link= Vinay Deodhar| journal = [[Enseign. Math.]] | mr = 0850554 | title=Some characterizations of Coxeter groups | volume = 32 | year =1986 | pages = 111–120 }}

Some characterizations of Coxeter groups | volume = 32 | year =1986 | pages = 111–120 }}

* {{cite journal|last=de Rham|first= G.|author-link=Georges de Rham|title= Sur les polygones générateurs de groupes fuchsiens| lang=fr |journal= [[Enseign. Math.]]|volume= 17 |year =1971|pages= 49–61|url= https://www.e-periodica.ch/cntmng?pid=ens-001%3A1971%3A17%3A%3A33}}

* {{cite book | last = Ellis | first = Graham | chapter = Triangle groups | isbn = 978-0-19-883298-0 | mr = 3971587 | pages = 441–444 | publisher = [[Oxford University Press]] | title = An Invitation to Computational Homotopy | year = 2019}}

* {{cite journal|mr = 1279064|last1=Epstein|first1= David B. A.|author1-link=David B. A. Epstein|last2= Petronio|first2= Carlo |title=An exposition of Poincaré's polyhedron theorem|journal=[[Enseign. Math.]]|volume= 40 | year= 1994|pages= 113–170|url= https://www.e-periodica.ch/cntmng?pid=ens-001%3A1994%3A40%3A%3A49}}

* {{citation |last=Evans |first=Ronald |title=A fundamental region for Hecke's modular group |journal=[[Journal of Number Theory]] |volume=5 |year=1973 |issue=2 |pages=108–115 |doi=10.1016/0022-314x(73)90063-2 |bibcode=1973JNT.....5..108E |doi-access=free }}

* {{citation |last1=Fricke |first1=Robert |author-link1=Robert Fricke|last2=Klein |first2=Felix |author-link2=Felix Klein |date=1897 |title=Vorlesungen über die Theorie der automorphen Functionen. Erster Band; Die gruppentheoretischen Grundlagen |url=https://archive.org/details/vorlesungenber01fricuoft |publisher=B. G. Teubner |language=de |isbn=978-1-4297-0551-6 |jfm=28.0334.01 }}

* {{citation |last1=Graham |first1=Ronald L. |author-link=Ronald Graham (mathematician) |last2=Knuth |first2=Donald E. |author-link2=Donald Knuth |last3=Patashnik |first3=Oren |author-link3=Oren Patashnik |year=1994 |title=Concrete mathematics |edition=2nd |publisher=Addison-Wesley |isbn=0-201-55802-5 |pages=116–118 }}

Line 543 ⟶ 534:

* {{citation |last=Helgason |first=Sigurdur |author-link=Sigurdur Helgason (mathematician) |title=Groups and geometric analysis. Integral geometry, invariant differential operators, and spherical functions |series=Mathematical Surveys and Monographs |volume=83 |publisher=American Mathematical Society |year=2000 |isbn=0-8218-2673-5 }}

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

* {{cite book|mr = 0649068| last= Hiller | first = Howard | title= Geometry of Coxeter groups| series=Research Notes in Mathematics| volume = 54 | publisher= Pitman | year = 1982 |isbn = 0-273-08517-4 }}

Research Notes in Mathematics| volume = 54 | publisher= Pitman | year = 1982 |isbn = 0-273-08517-4 }}

* {{cite book|first=Robert | last = Howlett| author-link= Robert Howlett| chapter= Introduction to Coxeter Groups| year= 1996 | chapter-url = https://www.maths.usyd.edu.au/u/ResearchReports/Algebra/How/1997-6.html |

title = Australian National University Geometric Group Theory Workshop|___location= [[Sydney]] }}

Line 550 ⟶ 540:

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

* {{citation |last=Iversen |first=Birger |title=Hyperbolic geometry |series=London Mathematical Society Student Texts |volume=25 |publisher=Cambridge University Press |year=1992 |isbn=0-521-43508-0 }}

* {{cite book| mr = 0217741 | last =Iwahori | first= Nagayoshi| author-link= Nagayoshi Iwahori|chapter =On discrete reflection groups on symmetric Riemannian manifolds| year = 1966| title= Proc. U.S.-Japan Seminar in Differential Geometry (Kyoto, 1965)| pages = 57–62| publisher = Nippon Hyoronsha| ___location= Tokyo}}

On discrete reflection groups on symmetric Riemannian manifolds| year = 1966| title= Proc. U.S.-Japan Seminar in Differential Geometry (Kyoto, 1965)| pages = 57–62| publisher = Nippon Hyoronsha| ___location= Tokyo}}

* {{cite book| mr = 3839612 | last =Johnson | first = Norman W. | title = Geometries and Transformations |

publisher = [[Cambridge University Press]]| year = 2018 | isbn = 978-1-107-10340-5 |

Line 557 ⟶ 546:

* {{citation |last=Magnus |first=Wilhelm |author-link=Wilhelm Magnus |title=Noneuclidean tesselations and their groups |series=Pure and Applied Mathematics |volume=61 |publisher=Academic Press |year=1974|url=

https://www.sciencedirect.com/bookseries/pure-and-applied-mathematics/vol/61/suppl/C}}

* {{cite book| mr = 0207802 | last1 = Magnus| first1 = Wilhelm| author1-link= Wilhelm Magnus| last2= Karrass| first2 = Abraham| last3 = Solitar | first3= Donald | author3-link = Donald Solitar | title=Combinatorial group theory: Presentations of groups in terms of generators and relations| year =1976|

Combinatorial group theory: Presentations of groups in terms of generators and relations| year =1976|

publisher= [[Dover Books]]| edition= Second revised}}

* {{citation |last=Maskit |first=Bernard |author-link=Bernard Maskit |title=On Poincaré's theorem for fundamental polygons |journal=[[Advances in Mathematics]] |volume=7 |pages=219–230 |year=1971 |issue=3 |doi=10.1016/s0001-8708(71)80003-8 |doi-access=free | url=https://www.sciencedirect.com/science/article/pii/S0001870871800038?via%3Dihub}}

* {{cite book | mr = 0959135 | last = Maskit| first = Bernard | author-link = Bernard Maskit| title=Kleinian groups| series = Grundlehren der mathematischen Wissenschaften | volume = 287| publisher= [[Springer-Verlag]] | year = 1988 | isbn = 3-540-17746-9| chapter = Poincaré's theorem}}

Kleinian groups| series = Grundlehren der mathematischen Wissenschaften | volume = 287| publisher= [[Springer-Verlag]] | year = 1988 | isbn = 3-540-17746-9| chapter = Poincaré's theorem}}

* {{cite journal| mr = 0679972 | last=Maxwell | first =George|

title = Sphere packings and hyperbolic reflection groups | journal = [[J. Algebra]]| volume = 79 | year=1982 | pages = 78–97| doi=10.1016/0021-8693(82)90318-0 }}

* {{citation |last=McMullen |first=Curtis T. |author-link=Curtis McMullen |title=Hausdorff dimension and conformal dynamics. III. Computation of dimension |journal=American Journal of Mathematics |volume=120 |year=1998 |pages=691–721 |doi=10.1353/ajm.1998.0031 |s2cid=15928775 }}

* {{cite journal|mr =0183818 | last= Matsumoto | first= Hideya| author-link= Hideya Matsumoto| title=Générateurs et relations des groupes de Weyl généralisés|lang = fr|journal=[[C. R. Acad. Sci. Paris]]|volume= 258 |year =1964 | pages= 3419–3422 | url= https://gallica.bnf.fr/ark:/12148/bpt6k4011c/f1029.item}}

Générateurs et relations des groupes de Weyl généralisés|lang = fr|journal=

[[C. R. Acad. Sci. Paris]]|volume= 258 |year =1964 | pages= 3419–3422 | url= https://gallica.bnf.fr/ark:/12148/bpt6k4011c/f1029.item}}

* {{cite book|mr=0418127|last= Milnor| first = John| author-link= John Milnor| chapter = On the 3-dimensional Brieskorn manifolds M(p,q,r)| title= Knots, groups, and 3-manifolds (Papers dedicated to the memory of R. H. Fox)|pages= 175–225|series= Ann. of Math. Studies| volume =84| publisher= [[Princeton University Press]] | year = 1975}}

* {{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 }}

* {{cite book| mr = 1299730| last = Ratcliffe |first = John G. | title =Foundations of hyperbolic manifolds| edition = Third| series = Graduate Texts in Mathematics | volume = 149

Foundations of hyperbolic manifolds| edition = Third| series = Graduate Texts in Mathematics | volume = 149

| publisher = Springer| year = 2019 | isbn = 978-3-030-31597-9}}

* {{cite book|mr = 3186310 |last= Schlag| first= Wilhelm |author-link = Wilhelm Schlag|title=A course in complex analysis and Riemann surfaces| series= Graduate Studies in Mathematics| volume = 154|

A course in complex analysis and Riemann surfaces| series= Graduate Studies in Mathematics| volume = 154|

publisher = [[American Mathematical Society]] | year= 2014 |isbn= 978-0-8218-9847-5|chapter= Uniformization

|pages = 305–351}}

Line 584 ⟶ 567:

* {{cite book |mr = 1333415 | last = Szenthe| first= J. | chapter= A significant property of real hyperbolic spaces| title= Proceedings of Symposium in Geometry (Cluj-Napoca and Tîrgu Mureş, 1992)| pages= 153–161| year = 1993}}

* {{citation |last=Takeuchi |first=Kisao |title=Arithmetic triangle groups |journal=Journal of the Mathematical Society of Japan |volume=29 |year=1977a |pages=91–106 |doi=10.2969/jmsj/02910091 |doi-access=free }}

* {{citation |last=Takeuchi |first=Kisao |title=Commensurability classes of arithmetic triangle groups |journal=Journal of the Faculty of Science, Thethe University of Tokyo, Section IA, Mathematics |volume=24 |year=1977b |pages=201–212 }}

* {{citation |last=Threlfall |first=W. |author-link=William Threlfall |url=http://digital.slub-dresden.de/fileadmin/data/304910686/304910686_tif/jpegs/304910686.pdf |title=Gruppenbilder |journal=Abhandlungen der Mathematisch-physischen Klasse der Sachsischen Akademie der Wissenschaften |volume=41 |pages=1–59 |publisher=Hirzel |year=1932 }}

* {{cite book|last=Tits|first= Jacques |author-link = Jacques Tits |title= Œuvres/Collected works, Volume I|lang=fr |editor1 = F. Buekenhout| editor2 = B. M. Mühlherr| editor3 = J-P. Tignol |editor4 = H. Van Maldeghem|series= Heritage of European Mathematics|url = https://www.ems-ph.org/books/book.php?proj_nr=168 |

Line 591 ⟶ 574:

* {{cite journal| first= Ernest B. | last = Vinberg| author-link = Ernest Vinberg| year=1971|

title = Discrete linear groups generated by reflections | journal = [[Mathematics of the USSR-Izvestiya]]|

pages= 1083–1119 | volume= 5 | issue = 5| doi = 10.1070/IM1971v005n05ABEH001203| bibcode = 1971IzMat...5.1083V| translator= P. Flor |mr = 0302779}}

* {{cite journal|first=Ernest B. |last = Vinberg| author-link = Ernest Vinberg|year= 1985|journal= [[Russian Mathematical Surveys]]|volume= 40|pages= 31–75 |url = https://iopscience.iop.org/article/10.1070/RM1985v040n01ABEH003527/pdf | publisher = [[London Mathematical Society]]| title = Hyperbolic reflection groups| issue=1 | doi=10.1070/RM1985v040n01ABEH003527 | bibcode=1985RuMaS..40...31V |translator = J. Wiegold}}

* {{cite book|mr=1254933|author1-link=Ernest Vinberg|last1=Vinberg|first1= Ernest B.|last2= Shvartsman|first2= O. V. | chapter= Discrete groups of motions of spaces of constant curvature|title = Geometry II: Spaces of Constant Curvature|pages= 139–248| series = Encyclopaedia Math. Sci.| volume = 29| publisher = [[Springer-Verlag]]|year= 1993| isbn = 3-540-52000-7}}

* {{citation |first=Matthias |last=Weber |title=Kepler's small stellated dodecahedron as a Riemann surface |journal=[[Pacific Journal of Mathematics]] |volume=220 |year=2005 |pages=167–182 |url=http://msp.org/pjm/2005/220-1/p09.xhtml |doi=10.2140/pjm.2005.220.167|doi-access=free }}