# 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 $\phi_{n}\to\phi$ in $\mathcal{D}(I)$, … 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. …
##### 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 $I_{\nu}\left(z\right)=\left(\tfrac{1}{2}z\right)^{\nu}\phi\left(1,\nu+1;\tfrac% {1}{4}z^{2}\right).$
For asymptotic expansions of $\phi\left(\rho,\beta;z\right)$ as $z\to\infty$ 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). …
##### 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 and an inverse hyperbolic function (or logarithm) if $x>y$: …
##### 9: 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. …
##### 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).