distributional completeness

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1: 14.30 Spherical and Spheroidal Harmonics

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Distributional Completeness

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2: 1.17 Integral and Series Representations of the Dirac Delta

… ►Equations (1.17.12_1) through (1.17.16) may re-interpreted as spectral representations of completeness relations, expressed in terms of Dirac delta distributions, as discussed in §1.18(v), and §1.18(vi) Further mathematical underpinnings are referenced in §1.17(iv). …

3: 29.3 Definitions and Basic Properties

… ►For each pair of values of $\nu$ and $k$ there are four infinite unbounded sets of real eigenvalues $h$ for which equation (29.2.1) has even or odd solutions with periods $2K$ or $4K$. … ►

§29.3(ii) Distribution

… ►In this table the nonnegative integer $m$ corresponds to the number of zeros of each Lamé function in $(0,K)$, whereas the superscripts $2m$, $2m+1$, or $2m+2$ correspond to the number of zeros in $[0,2K)$. … ►

${\displaystyle {\int }_0^K}\mathrm{dn}(x,k){\left({\mathrm{𝐸𝑐}}_{\nu }^{2m}(x,k^2)\right)}^{2}dx={\displaystyle \frac{1}{4}}\pi ,$

… ►To complete the definitions, ${\mathrm{𝐸𝑐}}_{\nu }^m(K,k^2)$ is positive and ${d{\mathrm{𝐸𝑠}}_{\nu }^m(z,k^2)/dz|}_{z=K}$ is negative. …

4: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►Such orthonormal sets are called complete. … ►of the Dirac delta distribution. … ►and completeness implies … ►and completeness relation … ►The formal completeness relation is now …

5: Errata

… ►The spectral theory of these operators, based on Sturm-Liouville and Liouville normal forms, distribution theory, is now discussed more completely, including linear algebra, matrices, matrices as linear operators, orthonormal expansions, Stieltjes integrals/measures, generating functions. …

6: 27.2 Functions

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27.2.1 $$n=\prod _{r=1}^{\nu \left(n\right)}{p}_r^{a_r},$$

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Symbols:

$\nu \left(n\right)$: number of distinct primes dividing a number, $n$: positive integer, $p,p_1,\mathrm{\dots }$: prime numbers and $a$: primitive root

Referenced by:

§27.2(i), §27.3

Permalink:

http://dlmf.nist.gov/27.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §27.2(i), §27.2 and Ch.27

… ►Tables of primes (§27.21) reveal great irregularity in their distribution. They tend to thin out among the large integers, but this thinning out is not completely regular. … ►

27.2.3 $$\pi \left(x\right)\sim \frac{x}{\ln x}.$$

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Symbols:

$\sim$: asymptotic equality, $\ln z$: principal branch of logarithm function, $\pi \left(x\right)$: number of primes not exceeding a number and $x$: real number

Referenced by:

§25.16(i), §27.11

Permalink:

http://dlmf.nist.gov/27.2.E3

Encodings:

pMML, png, TeX

Change of Notation (effective with 1.0.10):

The notation for logarithm has been changed to $\ln$ from $\log$.

See also:

Annotations for §27.2(i), §27.2 and Ch.27

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7: 22.4 Periods, Poles, and Zeros

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§22.4(i) Distribution

… ►Figure 22.4.1 illustrates the locations in the $z$-plane of the poles and zeros of the three principal Jacobian functions in the rectangle with vertices $0$, $2K$, $2K+2\mathrm{i}{K}^{\prime }$, $2\mathrm{i}{K}^{\prime }$. … ►For the distribution of the $k$-zeros of the Jacobian elliptic functions see Walker (2009). … ►Figure 22.4.2 depicts the fundamental unit cell in the $z$-plane, with vertices $\text{s}=0$, $\text{c}=K$, $\text{d}=K+\mathrm{i}{K}^{\prime }$, $\text{n}=\mathrm{i}{K}^{\prime }$. The set of points $z=mK+n\mathrm{i}{K}^{\prime }$, $m,n\in \mathbb{Z}$, comprise the lattice for the 12 Jacobian functions; all other lattice unit cells are generated by translation of the fundamental unit cell by $mK+n\mathrm{i}{K}^{\prime }$, where again $m,n\in \mathbb{Z}$. …

8: Bibliography J

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Tafeln höherer Funktionen (Tables of Higher Functions). 7th edition, B. G. Teubner, Stuttgart (Bilingual).

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Notes:

Reprint of sixth edition (McGraw-Hill, New York, 1960) with corrections and updated bibliography. The sixth edition is a completely revised and enlarged version of Jahnke and Emde (1945). In German and English.

External Links:

MathReview

Referenced by:

§19.1, §28.3(i).

Encodings:

BibTeX

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Distributions of matrix variates and latent roots derived from normal samples. Ann. Math. Statist. 35 (2), pp. 475–501.

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External Links:

FullText, MathReview (I. Olkin), ZentralBlatt

Referenced by:

§35.11, §35.4(i), §35.4(ii), §35.8(i), §35.8(ii), §35.8(iv), §35.9, Ch.35.

Encodings:

BibTeX

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Bounds on Dawson’s integral occurring in the analysis of a line distribution network for electric vehicles. Eurandom Preprint Series Technical Report 14, Eurandom, Eindhoven, The Netherlands.

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Notes:

ISSN 1389-2355

External Links:

FullText

Referenced by:

§7.8, §7.8.

Encodings:

BibTeX

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Continuous Univariate Distributions. 2nd edition, Vol. I, John Wiley & Sons Inc., New York.

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External Links:

ISBN 0-471-58495-9, MathReview (D. N. Shanbhag), ZentralBlatt

Referenced by:

§7.20(iii), §8.23.

Encodings:

BibTeX

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Continuous Univariate Distributions. 2nd edition, Vol. II, John Wiley & Sons Inc., New York.

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External Links:

ISBN 0-471-58494-0, MathReview (D. N. Shanbhag), ZentralBlatt

Referenced by:

§8.23.

Encodings:

BibTeX

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9: 2.6 Distributional Methods

§2.6 Distributional Methods

… ►Motivated by the definition of distributional derivatives, we can assign them the distributions defined by … ►The Dirac delta distribution in (2.6.17) is given by … ►These equations again hold only in the sense of distributions. … ►

§2.6(iv) Regularization

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10: Philip J. Davis

… ►He returned to Harvard after the war and completed a Ph. … ►He also had a big influence on the development of the NBS Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (A&S), which became one of the most widely distributed and highly cited publications in NIST’s history. … ►Decades later, Olver became Editor-in-Chief and Mathematics Editor of the NIST Digital Library of Mathematical Functions (DLMF), a complete revision of A&S that was publicly released in 2010. …