generalized functions

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1: 1.16 Distributions

… ►

\Lambda:\mathcal{D}(I)\rightarrow\mathbb{C}

is called a distribution, or generalized function, if it is a continuous linear functional on

\mathcal{D}(I)

, that is, it is a linear functional and for every

\phi_{n}\to\phi

\mathcal{D}(I)

, … ►More generally, for

\alpha\colon[a,b]\to[-\infty,\infty]

a nondecreasing function the corresponding Lebesgue–Stieltjes measure

\mu_{\alpha}

(see §1.4(v)) can be considered as a distribution: … ►More generally, if

\alpha(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

\phi\left(\rho,\beta;z\right)

is defined by … ►

10.46.2

I_{\nu}\left(z\right)=\left(\tfrac{1}{2}z\right)^{\nu}\phi\left(1,\nu+1;\tfrac% {1}{4}z^{2}\right).

ⓘ

Symbols:: $\phi\left(\NVar{\rho},\NVar{\beta};\NVar{z}\right)$ : generalized Bessel function, $I_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the first kind, $z$ : complex variable and $\nu$ : complex parameter
Permalink:: http://dlmf.nist.gov/10.46.E2
Encodings:: pMML, png, TeX
See also:: Annotations for §10.46 and Ch.10

►For asymptotic expansions of

\phi\left(\rho,\beta;z\right)

z\to\infty

in various sectors of the complex

z

-plane for fixed real values of

\rho

and fixed real or complex values of

\beta

, see Wright (1935) when

\rho>0

, and Wright (1940b) when

-1<\rho<0

. … ►The Laplace transform of

\phi\left(\rho,\beta;z\right)

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

\int_{0}^{x}e^{-t^{p}}\,\mathrm{d}t

and

\int_{0}^{x}e^{t^{p}}\,\mathrm{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: $R_{C}\left(x,y\right)$

… ►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}\left(x,y\right)

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) $\mathit{3j}$ , $\mathit{6j}$ , and $\mathit{9j}$ 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).