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31: 35.1 Special Notation
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►(For other notation see Notation for the Special Functions.)
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►The main functions treated in this chapter are the multivariate gamma and beta functions, respectively and , and the special functions of matrix argument: Bessel (of the first kind) and (of the second kind) ; confluent hypergeometric (of the first kind) or and (of the second kind) ; Gaussian hypergeometric or ; generalized hypergeometric or .
►An alternative notation for the multivariate gamma function is (Herz (1955, p. 480)).
Related notations for the Bessel functions are (Faraut and Korányi (1994, pp. 320–329)), (Terras (1988, pp. 49–64)), and (Faraut and Korányi (1994, pp. 357–358)).
complex variables. | |
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32: 30.11 Radial Spheroidal Wave Functions
§30.11 Radial Spheroidal Wave Functions
►§30.11(i) Definitions
… ►Connection Formulas
… ►§30.11(ii) Graphics
… ►§30.11(iv) Wronskian
…33: 28.12 Definitions and Basic Properties
§28.12 Definitions and Basic Properties
… ►§28.12(ii) Eigenfunctions
… ►If is a normal value of the corresponding equation (28.2.16), then these functions are uniquely determined as analytic functions of and by the normalization … ► … ►34: 1.16 Distributions
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§1.16(i) Test Functions
… ► … ► … ► is called a distribution, or generalized function, if it is a continuous linear functional on , that is, it is a linear functional and for every in , … ►§1.16(iv) Heaviside Function
…35: 16.17 Definition
§16.17 Definition
… ►Then the Meijer -function is defined via the Mellin–Barnes integral representation: … ► ►When more than one of Cases (i), (ii), and (iii) is applicable the same value is obtained for the Meijer -function. … ►Then …36: 35.5 Bessel Functions of Matrix Argument
§35.5 Bessel Functions of Matrix Argument
►§35.5(i) Definitions
… ►§35.5(ii) Properties
… ►§35.5(iii) Asymptotic Approximations
►For asymptotic approximations for Bessel functions of matrix argument, see Herz (1955) and Butler and Wood (2003).37: 22.2 Definitions
§22.2 Definitions
… ►As a function of , with fixed , each of the 12 Jacobian elliptic functions is doubly periodic, having two periods whose ratio is not real. … … ►The Jacobian functions are related in the following way. … ►In terms of Neville’s theta functions (§20.1) …38: 21.2 Definitions
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