About the Project

generalized functions

AdvancedHelp

(0.006 seconds)

1—10 of 376 matching pages

1: 1.16 Distributions
Λ : 𝒟 ( I ) is called a distribution, or generalized function, if it is a continuous linear functional on 𝒟 ( I ) , that is, it is a linear functional and for every ϕ n ϕ in 𝒟 ( I ) , … More generally, for α : [ a , b ] [ , ] a nondecreasing function the corresponding Lebesgue–Stieltjes measure μ α (see §1.4(v)) can be considered as a distribution: … More generally, if α ( x ) is an infinitely differentiable function, then … Friedman (1990) gives an overview of generalized functions and their relation to distributions. …
2: 16.2 Definition and Analytic Properties
§16.2(i) Generalized Hypergeometric Series
Unless indicated otherwise it is assumed that in the DLMF generalized hypergeometric functions assume their principal values. …
Polynomials
§16.2(v) Behavior with Respect to Parameters
3: 35.8 Generalized Hypergeometric Functions of Matrix Argument
§35.8 Generalized Hypergeometric Functions of Matrix Argument
§35.8(i) Definition
Convergence Properties
§35.8(ii) Relations to Other Functions
Confluence
4: 10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function
§10.46 Generalized and Incomplete Bessel Functions; Mittag-Leffler Function
The function ϕ ( ρ , β ; z ) is defined by …
10.46.2 I ν ( z ) = ( 1 2 z ) ν ϕ ( 1 , ν + 1 ; 1 4 z 2 ) .
For asymptotic expansions of ϕ ( ρ , β ; z ) as z in various sectors of the complex z -plane for fixed real values of ρ and fixed real or complex values of β , see Wright (1935) when ρ > 0 , and Wright (1940b) when 1 < ρ < 0 . … The Laplace transform of ϕ ( ρ , β ; z ) can be expressed in terms of the Mittag-Leffler function: …
5: 8.16 Generalizations
§8.16 Generalizations
For a generalization of the incomplete gamma function, including asymptotic approximations, see Chaudhry and Zubair (1994, 2001) and Chaudhry et al. (1996). …
6: 17.15 Generalizations
§17.15 Generalizations
7: 7.16 Generalized Error Functions
§7.16 Generalized Error Functions
Generalizations of the error function and Dawson’s integral are 0 x e t p d t and 0 x e t p d t . …
8: 19.2 Definitions
§19.2(i) General Elliptic Integrals
Let s 2 ( t ) be a cubic or quartic polynomial in t with simple zeros, and let r ( s , t ) be a rational function of s and t containing at least one odd power of s . …
§19.2(iv) A Related Function: R C ( x , y )
In (19.2.18)–(19.2.22) the inverse trigonometric and hyperbolic functions assume their principal values (§§4.23(ii) and 4.37(ii)). When x and y are positive, R C ( x , y ) is an inverse circular function if x < y and an inverse hyperbolic function (or logarithm) if x > y : …
9: 16.24 Physical Applications
§16.24 Physical Applications
§16.24(i) Random Walks
Generalized hypergeometric functions and Appell functions appear in the evaluation of the so-called Watson integrals which characterize the simplest possible lattice walks. …
§16.24(iii) 3 j , 6 j , and 9 j Symbols
10: 16.26 Approximations
§16.26 Approximations
For discussions of the approximation of generalized hypergeometric functions and the Meijer G -function in terms of polynomials, rational functions, and Chebyshev polynomials see Luke (1975, §§5.12 - 5.13) and Luke (1977b, Chapters 1 and 9).