Horn Hypergeometric Series

	Horn Hypergeometric Series In the theory of special functions in mathematics, the Horn functions (named for Jakob Horn) are the 34 distinct convergent hypergeometric series In mathematics, the Gaussian or ordinary hypergeometric function 2''F''1(''a'',''b'';''c'';''z'') is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is ... of order two (i.e. having two independent variables), enumerated by (corrected by ). They are listed in . B. C. Carlson revealed a problem with the Horn function classification scheme. The total 34 Horn functions can be further categorised into 14 complete hypergeometric functions and 20 confluent hypergeometric functions. The complete functions, with their domain of convergence, are: * F_1(\alpha;\beta,\beta';\gamma;z,w)\equiv\sum_^\sum_^\frac\frac/;, z, <1\land, w, <1 * $F_2(\alpha;\beta,\beta';\gamma,\gamma';z,w)\equiv\sum_^\sum_^\frac\frac/;, z, +, w, <1$ * $...$ [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Special Function Special functions are particular mathematical functions that have more or less established names and notations due to their importance in mathematical analysis, functional analysis, geometry, physics, or other applications. The term is defined by consensus, and thus lacks a general formal definition, but the list of mathematical functions contains functions that are commonly accepted as special. Tables of special functions Many special functions appear as solutions of differential equations or integrals of elementary functions. Therefore, tables of integrals usually include descriptions of special functions, and tables of special functions include most important integrals; at least, the integral representation of special functions. Because symmetries of differential equations are essential to both physics and mathematics, the theory of special functions is closely related to the theory of Lie groups and Lie algebras, as well as certain topics in mathematical physics. Symbolic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
picture info	Mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Jakob Horn Jakob Horn (14 February 1867 – 24 February 1946) was a German mathematician who introduced Horn function In the theory of special functions in mathematics, the Horn functions (named for Jakob Horn) are the 34 distinct convergent hypergeometric series In mathematics, the Gaussian or ordinary hypergeometric function 2''F''1(''a'',''b'';''c'';''z'') ...s. Works * (123 pages) * (59 pages) * * * * (237 pages) References External links Jahrbuch für Mathematik (search on author: horn, j) 1867 births 19th-century German mathematicians 1946 deaths 20th-century German mathematicians {{Germany-mathematician-stub ... [...More Info...] [...Related Items...] OR:* [Wikipedia] [Google] [Baidu]
	Hypergeometric Series In mathematics, the Gaussian or ordinary hypergeometric function 2''F''1(''a'',''b'';''c'';''z'') is a special function represented by the hypergeometric series, that includes many other special functions as specific or limiting cases. It is a solution of a second-order linear ordinary differential equation (ODE). Every second-order linear ODE with three regular singular points can be transformed into this equation. For systematic lists of some of the many thousands of published identities involving the hypergeometric function, see the reference works by and . There is no known system for organizing all of the identities; indeed, there is no known algorithm that can generate all identities; a number of different algorithms are known that generate different series of identities. The theory of the algorithmic discovery of identities remains an active research topic. History The term "hypergeometric series" was first used by John Wallis in his 1655 book ''Arithmetica Infinitor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]