Kampé De Fériet Function

	Kampé De Fériet Function In mathematics, the Kampé de Fériet function is a two-variable generalization of the generalized hypergeometric series, introduced by Joseph Kampé de Fériet. The Kampé de Fériet function is given by : ^F_\left( \begin a_1,\cdots,a_p\colon b_1,b_1';\cdots;b_q,b_q'; \\ c_1,\cdots,c_r\colon d_1,d_1';\cdots;d_s,d_s'; \end x,y\right)= \sum_^\infty\sum_^\infty\frac\frac\cdot\frac. Applications The general sextic equation can be solved in terms of Kampé de Fériet functions. See also Appell series Humbert series In mathematics, Humbert series are a set of seven hypergeometric series Φ1, Φ2, Φ3, Ψ1, Ψ2, Ξ1, Ξ2 of two variables that generalize Kummer's confluent hypergeometric series 1''F''1 of one variable and the confluent hypergeometric limit fu ... * Lauricella series (three-variable) References * * * External links * Hypergeometric functions {{analysis-stub ... [...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]
	Generalized Hypergeometric Function In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by ''n'' is a rational function of ''n''. The series, if convergent, defines a generalized hypergeometric function, which may then be defined over a wider domain of the argument by analytic continuation. The generalized hypergeometric series is sometimes just called the hypergeometric series, though this term also sometimes just refers to the Gaussian hypergeometric series. Generalized hypergeometric functions include the (Gaussian) hypergeometric function and the confluent hypergeometric function as special cases, which in turn have many particular special functions as special cases, such as elementary functions, Bessel functions, and the orthogonal polynomials, classical orthogonal polynomials. Notation A hypergeometric series is formally defined as a power series :\beta_0 + \beta_1 z + \beta_2 z^2 + \dots = \sum_ \beta_n z^n in which the ratio of succe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Joseph Kampé De Fériet Marie-Joseph Kampé de Fériet (14 May 1893 – 6 April 1982) was a French mathematician at Université Lille Nord de France from 1919 to 1969. Besides his works on mathematics and fluid mechanics, he directed the ''Institut de mécanique des fluides de Lille'' ( ONERA Lille) and taught fluid dynamics and information theory at École centrale de Lille from 1930 to 1969. He devised the Kampé de Fériet functions, which further generalize the generalized hypergeometric function In mathematics, a generalized hypergeometric series is a power series in which the ratio of successive coefficients indexed by ''n'' is a rational function of ''n''. The series, if convergent, defines a generalized hypergeometric function, which ...s. He was an Invited Speaker of the ICM in 1928 at Bologna, in 1932 at Zurich, and in 1954 at Amsterdam. Works * J. Kampé de Fériet & P.E. Appell ''Fonctions hypergéometriques et hypersphériques'' (Paris, Gauthier-Villars, 1926) * J. Kampé de Fériet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Sextic Equation In algebra, a sextic (or hexic) polynomial is a polynomial of degree six. A sextic equation is a polynomial equation of degree six—that is, an equation whose left hand side is a sextic polynomial and whose right hand side is zero. More precisely, it has the form: :ax^6+bx^5+cx^4+dx^3+ex^2+fx+g=0,\, where and the ''coefficients'' may be integers, rational numbers, real numbers, complex numbers or, more generally, members of any field. A sextic function is a function defined by a sextic polynomial. Because they have an even degree, sextic functions appear similar to quartic functions when graphed, except they may possess an additional local maximum and local minimum each. The derivative of a sextic function is a quintic function. Since a sextic function is defined by a polynomial with even degree, it has the same infinite limit when the argument goes to positive or negative infinity. If the leading coefficient is positive, then the function increases to positive infinity ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Appell Series In mathematics, Appell series (mathematics), series are a set of four hypergeometric series ''F''1, ''F''2, ''F''3, ''F''4 of two variable (mathematics), variables that were introduced by and that generalize hypergeometric function, Gauss's hypergeometric series 2''F''1 of one variable. Appell established the set of partial differential equations of which these function (mathematics), functions are solutions, and found various reduction formulas and expressions of these series in terms of hypergeometric series of one variable. Definitions The Appell series ''F''1 is defined for , ''x'', < 1, , ''y'', < 1 by the double series : $F_1(a,b_1,b_2;c;x,y) = \sum_^\infty \frac \,x^m y^n ~,$ where $(q)_n$ is the rising factorial Pochhammer symbol. For other values of ''x'' and ''y'' the function ''F''₁ can be defined by analytic continuation. It can be shown that : $F_1(a,b_1,b_2;c;x,y) = \sum_^\infty \frac \,x^r y^r _2F_\left(a+r,b_1+r;c+2r;x\r ... [...More Info...] [...Related Items...]$ OR: [Wikipedia] [Google] [Baidu]
	Humbert Series In mathematics, Humbert series are a set of seven hypergeometric series Φ1, Φ2, Φ3, Ψ1, Ψ2, Ξ1, Ξ2 of two variables that generalize Kummer's confluent hypergeometric series 1''F''1 of one variable and the confluent hypergeometric limit function 0''F''1 of one variable. The first of these double series was introduced by . Definitions The Humbert series Φ1 is defined for , ''x'', \real \,a > 0 ~. This representation can be verified by means of Taylor expansion of the integrand, followed by termwise integration. Similarly, the function Φ2 is defined for all ''x'', ''y'' by the series: : \Phi_2(b_1,b_2,c;x,y) = F_1(-,b_1,b_2,c;x,y) = \sum_^\infty \frac \,x^m y^n ~, the function Φ3 for all ''x'', ''y'' by the series: : \Phi_3(b,c;x,y) = \Phi_2(b,-,c;x,y) = F_1(-,b,-,c;x,y) = \sum_^\infty \frac \,x^m y^n ~, the function Ψ1 for , ''x'', < 1 by the series: : $\Psi_1(a,b,c_1,c_2;x,y) = F_2(a,b,-,c_1,c_2;x,y) = \sum_^\infty \frac \,x^m y^n ~,$ the f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]
	Lauricella Hypergeometric Series In 1893 Giuseppe Lauricella defined and studied four hypergeometric series ''F''''A'', ''F''''B'', ''F''''C'', ''F''''D'' of three variables. They are : : F_A^(a,b_1,b_2,b_3,c_1,c_2,c_3;x_1,x_2,x_3) = \sum_^ \frac \,x_1^x_2^x_3^ for , ''x''1, + , ''x''2, + , ''x''3, < 1 and : $F_B^(a_1,a_2,a_3,b_1,b_2,b_3,c;x_1,x_2,x_3) = \sum_^ \frac \,x_1^x_2^x_3^$ for , ''x''₁, < 1, , ''x''₂, < 1, , ''x''₃, < 1 and : $F_C^(a,b,c_1,c_2,c_3;x_1,x_2,x_3) = \sum_^ \frac \,x_1^x_2^x_3^$ for , ''x''₁, ^1/2 + , ''x''₂, ^1/2 + , ''x''₃, ^1/2 < 1 and : $F_D^(a,b_1,b_2,b_3,c;x_1,x_2,x_3) = \sum_^ \frac \,x_1^x_2^x_3^$ for , ''x''₁, < 1, , ''x''₂, < 1, , ''x''₃, < 1. Here the Pochhammer symbol [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu]