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21: 1.16 Distributions

… ► $\mathrm{\Lambda }:𝒟(I)\to ℂ$ 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 ${\varphi }_n\to \varphi$ in $𝒟(I)$, … ►A tempered distribution is a continuous linear functional $\mathrm{\Lambda }$ on $𝒯$. … ►A distribution in ${ℝ}^n$ is a continuous linear functional on ${𝒟}_n$. … ►Tempered distributions are continuous linear functionals on this space of test functions. …

22: Bibliography E

… ►

Generating functions of certain continuous orthogonal systems. Proc. Roy. Soc. Edinburgh. Sect. A. 61, pp. 61–70.

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External Links:

MathReview (G. Szegő), ZentralBlatt

Referenced by:

§12.16.

Encodings:

BibTeX

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23: 10.18 Modulus and Phase Functions

… ►where $M_{\nu }\left(x\right)$ $(>0)$, $N_{\nu }\left(x\right)$ $(>0)$, ${\theta }_{\nu }\left(x\right)$, and ${\varphi }_{\nu }\left(x\right)$ are continuous real functions of $\nu$ and $x$, with the branches of ${\theta }_{\nu }\left(x\right)$ and ${\varphi }_{\nu }\left(x\right)$ fixed by …

24: 10.68 Modulus and Phase Functions

… ►where $M_{\nu }\left(x\right)(>0)$, $N_{\nu }\left(x\right)(>0)$, ${\theta }_{\nu }\left(x\right)$, and ${\varphi }_{\nu }\left(x\right)$ are continuous real functions of $x$ and $\nu$, with the branches of ${\theta }_{\nu }\left(x\right)$ and ${\varphi }_{\nu }\left(x\right)$ chosen to satisfy (10.68.18) and (10.68.21) as $x\to \mathrm{\infty }$. …

25: 18.39 Applications in the Physical Sciences

… ►The spectrum is mixed as in §1.18(viii), with the discrete eigenvalues given by (18.39.18) and the continuous eigenvalues by ${(\alpha \mathrm{\hslash }\gamma )}^2/(2m)$ ($\gamma \ge 0$) with corresponding eigenfunctions ${\mathrm{e}}^{\alpha (x-x_e)/2}{W}_{\lambda ,\mathrm{i}\gamma }\left(2\lambda {\mathrm{e}}^{-\alpha (x-x_e)}\right)$ expressed in terms of Whittaker functions (13.14.3). … ►Given that $a=-b$ in both the attractive and repulsive cases, the expression for the absolutely continuous, $x\in [-1,1]$, part of the function of (18.35.6) may be simplified: ►

18.39.50 $$w^{\mathrm{CP}}(x)=\frac{(l+1+\frac{2Z}{s})}{\pi \mathrm{\Gamma }(2l+2)}{\mathrm{e}}^{(2\theta (x)-\pi )\tau (x)}{\left(4(1-x^2)\right)}^{l+\frac{1}{2}}{\left|\mathrm{\Gamma }\left(l+1+\mathrm{i}\tau (x)\right)\right|}^2,$$ $\theta (x)=\arccos (x)$, $\tau (x)=-\frac{2Z}{s}\sqrt{\frac{1-x}{1+x}}$.

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\arccos z$: arccosine function, $w(x)$: weight function, $l$: nonnegative integer and $x$: real variable

Referenced by:

Figure 18.39.2, Figure 18.39.2, §18.39(iv), §18.40(ii)

Permalink:

http://dlmf.nist.gov/18.39.E50

Encodings:

pMML, png, TeX

See also:

Annotations for §18.39(iv), §18.39(iv), §18.39 and Ch.18

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26: 22.14 Integrals

… ►The branches of the inverse trigonometric functions are chosen so that they are continuous. … ►Again, the branches of the inverse trigonometric functions must be continuous. …

27: 28.29 Definitions and Basic Properties

… ► $Q(z)$ is either a continuous and real-valued function for $z\in ℝ$ or an analytic function of $z$ in a doubly-infinite open strip that contains the real axis. …

28: 2.6 Distributional Methods

… ►, a continuous linear functional) on the space $𝒯$ of rapidly decreasing functions on $ℝ$. … ►Since the function $t^{\mu }\left(\ln t-\gamma -\psi \left(\mu +1\right)\right)$ and all its derivatives are locally absolutely continuous in $(0,\mathrm{\infty })$, the distributional derivatives in the first sum in (2.6.44) can be replaced by the corresponding ordinary derivatives. …

29: 18.27 $q$-Hahn Class

… ►Thus in addition to a relation of the form (18.27.2), such systems may also satisfy orthogonality relations with respect to a continuous weight function on some interval. …

30: 24.17 Mathematical Applications

… ►Let ${𝒮}_n$ denote the class of functions that have $n-1$ continuous derivatives on $ℝ$ and are polynomials of degree at most $n$ in each interval $(k,k+1)$, $k\in \mathbb{Z}$. …