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In the mathematical theory of special functions, Schwarz's list or the Schwartz table is the list of 15 cases found by Hermann Schwarz when hypergeometric functions can be expressed algebraically. More precisely, it is a listing of parameters determining the cases in which the hypergeometric equation has a finite monodromy group, or equivalently has two independent solutions that are algebraic functions. It lists 15 cases, divided up by the isomorphism class of the monodromy group (excluding the case of a cyclic group), and was first derived by Schwarz by methods of complex analytic geometry. Correspondingly the statement is not directly in terms of the parameters specifying the hypergeometric equation, but in terms of quantities used to describe certain spherical triangles.

Attributes	Values
rdf:type	yago:Abstraction100002137 yago:Function113783816 yago:MathematicalRelation113783581 yago:Relation100031921 yago:WikicatHypergeometricFunctions
rdfs:label	Schwarz's list (en)
rdfs:comment	In the mathematical theory of special functions, Schwarz's list or the Schwartz table is the list of 15 cases found by Hermann Schwarz when hypergeometric functions can be expressed algebraically. More precisely, it is a listing of parameters determining the cases in which the hypergeometric equation has a finite monodromy group, or equivalently has two independent solutions that are algebraic functions. It lists 15 cases, divided up by the isomorphism class of the monodromy group (excluding the case of a cyclic group), and was first derived by Schwarz by methods of complex analytic geometry. Correspondingly the statement is not directly in terms of the parameters specifying the hypergeometric equation, but in terms of quantities used to describe certain spherical triangles. (en)
dcterms:subject	Hypergeometric functions
Wikipage page ID	26956489 (xsd:integer)
Wikipage revision ID	1102216277 (xsd:integer)
Link from a Wikipage to another Wikipage	Schwarz triangle Projective unitary group Algebraic function Hypergeometric functions Hypergeometric function Riemann surface Hypergeometric differential equation Coxeter–Dynkin diagram Mathematics Generalized hypergeometric function George Mostow Arithmetic group Zariski closure Émile Picard Identity component Cyclic group Felix Klein Numerical integration Discrete group Riemann sphere Triangle group Pierre Deligne Special function Spherical triangle Michio Kuga Solvable group Lamé equation Hypergeometric equation Journal für die reine und angewandte Mathematik Regular singularities Lattice (group theory) Differential Galois group Monodromy group
Link from a Wikipage to an external page	http://www.math.ens.fr/~boalch/files/nlsl.pdf http://resolver.sub.uni-goettingen.de/purl%3FGDZPPN002155206 http://repository.kulib.kyoto-u.ac.jp/dspace/bitstream/2433/84920/1/Lectures_in_Mathematics_15.pdf
sameAs	Schwarz's list Schwarz's list Schwarz's list Schwarz's list
dbp:wikiPageUsesTemplate	dbt:Cite_book dbt:Cite_journal dbt:Reflist dbt:Technical dbt:MathSciNet dbt:Harvs
authorlink	Hermann Schwarz (en)
first	Hermann (en)
last	Schwarz (en)
year	1873 (xsd:integer)
loc	p. 323 (en)
has abstract	In the mathematical theory of special functions, Schwarz's list or the Schwartz table is the list of 15 cases found by Hermann Schwarz when hypergeometric functions can be expressed algebraically. More precisely, it is a listing of parameters determining the cases in which the hypergeometric equation has a finite monodromy group, or equivalently has two independent solutions that are algebraic functions. It lists 15 cases, divided up by the isomorphism class of the monodromy group (excluding the case of a cyclic group), and was first derived by Schwarz by methods of complex analytic geometry. Correspondingly the statement is not directly in terms of the parameters specifying the hypergeometric equation, but in terms of quantities used to describe certain spherical triangles. The wider importance of the table, for general second-order differential equations in the complex plane, was shown by Felix Klein, who proved a result to the effect that cases of finite monodromy for such equations and regular singularities could be attributed to changes of variable (complex analytic mappings of the Riemann sphere to itself) that reduce the equation to hypergeometric form. In fact more is true: Schwarz's list underlies all second-order equations with regular singularities on compact Riemann surfaces having finite monodromy, by a pullback from the hypergeometric equation on the Riemann sphere by a complex analytic mapping, of degree computable from the equation's data. The numbers are (up to permutations, sign changes and addition of with even) the differences of the exponents of the hypergeometric differential equation at the three singular points . They are rational numbers if and only if and are, a point that matters in arithmetic rather than geometric approaches to the theory. (en)
gold:hypernym	List
prov:wasDerivedFrom	wikipedia-en:Schwarz's_list?oldid=1102216277&ns=0
page length (characters) of wiki page	8631 (xsd:nonNegativeInteger)
foaf:isPrimaryTopicOf	wikipedia-en:Schwarz's_list
is Link from a Wikipage to another Wikipage of	Hypergeometric function Hermann Schwarz Grothendieck–Katz p-curvature conjecture Schwartz's list Schwartz table Schwarz list Schwarz table
is Wikipage redirect of	Schwartz's list Schwartz table Schwarz list Schwarz table
is foaf:primaryTopic of	wikipedia-en:Schwarz's_list

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