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- Schwarz's_list 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 (1873, p. 323) 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.".
- Schwarz's_list wikiPageExternalLink Lectures_in_Mathematics_15.pdf.
- Schwarz's_list wikiPageExternalLink purl?GDZPPN002155206.
- Schwarz's_list wikiPageExternalLink nlsl.pdf.
- Schwarz's_list wikiPageID "26956489".
- Schwarz's_list wikiPageRevisionID "604574835".
- Schwarz's_list authorlink "Hermann Schwarz".
- Schwarz's_list first "Hermann".
- Schwarz's_list hasPhotoCollection Schwarz's_list.
- Schwarz's_list last "Schwarz".
- Schwarz's_list loc "p. 323".
- Schwarz's_list year "1873".
- Schwarz's_list subject Category:Hypergeometric_functions.
- Schwarz's_list type Abstraction100002137.
- Schwarz's_list type Function113783816.
- Schwarz's_list type HypergeometricFunctions.
- Schwarz's_list type MathematicalRelation113783581.
- Schwarz's_list type Relation100031921.
- Schwarz's_list 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 (1873, p. 323) 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.".
- Schwarz's_list label "Schwarz's list".
- Schwarz's_list sameAs m.0bs1x9_.
- Schwarz's_list sameAs Q7433183.
- Schwarz's_list sameAs Q7433183.
- Schwarz's_list sameAs Schwarz's_list.
- Schwarz's_list wasDerivedFrom Schwarz's_list?oldid=604574835.
- Schwarz's_list isPrimaryTopicOf Schwarz's_list.