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- Schwarzs_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.The numbers λ, μ, ν are (up to a sign) the differences 1 − c, c − a − b, a − b of the exponents of the hypergeometric differential equation at the three singular points 0, 1, ∞. They are rational numbers if and only if a, b and c are, a point that matters in arithmetic rather than geometric approaches to the theory.".
- Schwarzs_list wikiPageExternalLink Lectures_in_Mathematics_15.pdf.
- Schwarzs_list wikiPageExternalLink purl?GDZPPN002155206.
- Schwarzs_list wikiPageExternalLink nlsl.pdf.
- Schwarzs_list wikiPageID "26956489".
- Schwarzs_list wikiPageLength "6690".
- Schwarzs_list wikiPageOutDegree "32".
- Schwarzs_list wikiPageRevisionID "670167709".
- Schwarzs_list wikiPageWikiLink Algebraic_function.
- Schwarzs_list wikiPageWikiLink Arithmetic_group.
- Schwarzs_list wikiPageWikiLink Category:Hypergeometric_functions.
- Schwarzs_list wikiPageWikiLink Coxeter–Dynkin_diagram.
- Schwarzs_list wikiPageWikiLink Crelles_Journal.
- Schwarzs_list wikiPageWikiLink Cyclic_group.
- Schwarzs_list wikiPageWikiLink Differential_Galois_theory.
- Schwarzs_list wikiPageWikiLink Discrete_group.
- Schwarzs_list wikiPageWikiLink Felix_Klein.
- Schwarzs_list wikiPageWikiLink Generalized_hypergeometric_function.
- Schwarzs_list wikiPageWikiLink George_Mostow.
- Schwarzs_list wikiPageWikiLink Hypergeometric_function.
- Schwarzs_list wikiPageWikiLink Identity_component.
- Schwarzs_list wikiPageWikiLink Lamé_function.
- Schwarzs_list wikiPageWikiLink Lattice_(discrete_subgroup).
- Schwarzs_list wikiPageWikiLink Michio_Kuga.
- Schwarzs_list wikiPageWikiLink Monodromy.
- Schwarzs_list wikiPageWikiLink Numerical_integration.
- Schwarzs_list wikiPageWikiLink Pierre_Deligne.
- Schwarzs_list wikiPageWikiLink Projective_unitary_group.
- Schwarzs_list wikiPageWikiLink Regular_singular_point.
- Schwarzs_list wikiPageWikiLink Riemann_sphere.
- Schwarzs_list wikiPageWikiLink Riemann_surface.
- Schwarzs_list wikiPageWikiLink Schwarz_triangle.
- Schwarzs_list wikiPageWikiLink Solvable_group.
- Schwarzs_list wikiPageWikiLink Special_functions.
- Schwarzs_list wikiPageWikiLink Spherical_trigonometry.
- Schwarzs_list wikiPageWikiLink Triangle_group.
- Schwarzs_list wikiPageWikiLink Zariski_topology.
- Schwarzs_list wikiPageWikiLink Émile_Picard.
- Schwarzs_list wikiPageWikiLinkText "Schwarz's list ".
- Schwarzs_list wikiPageWikiLinkText "Schwarz's list".
- Schwarzs_list authorlink "Hermann Schwarz".
- Schwarzs_list first "Hermann".
- Schwarzs_list last "Schwarz".
- Schwarzs_list loc "p. 323".
- Schwarzs_list wikiPageUsesTemplate Template:Bots.
- Schwarzs_list wikiPageUsesTemplate Template:Citation.
- Schwarzs_list wikiPageUsesTemplate Template:Harvs.
- Schwarzs_list wikiPageUsesTemplate Template:MathSciNet.
- Schwarzs_list wikiPageUsesTemplate Template:Reflist.
- Schwarzs_list year "1873".
- Schwarzs_list subject Category:Hypergeometric_functions.
- Schwarzs_list hypernym List.
- Schwarzs_list type Type.
- Schwarzs_list type Function.
- Schwarzs_list type Type.
- Schwarzs_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.".
- Schwarzs_list label "Schwarz's list".
- Schwarzs_list sameAs Q7433183.
- Schwarzs_list sameAs m.0bs1x9_.
- Schwarzs_list sameAs Q7433183.
- Schwarzs_list wasDerivedFrom Schwarzs_list?oldid=670167709.
- Schwarzs_list isPrimaryTopicOf Schwarzs_list.