distributional completeness

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11: DLMF Project News

error generating summary

12: 28.12 Definitions and Basic Properties

… ► … ►To complete the definition we require … ►To complete the definitions of the ${\mathrm{me}}_{\nu }$ functions we set …

13: 1.16 Distributions

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1.16.30 $$𝐃=(\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_1},\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_2},\mathrm{\dots },\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_n}).$$

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Defines:

$𝐃$: vector differential operator (locally)

Symbols:

$\mathrm{i}$: imaginary unit, $\frac{\partial f}{\partial x}$: partial derivative of $f$ with respect to $x$ , $\partial x$: partial differential of $x$ , $n$: nonnegative integer, $𝐷f$: distributional derivative and $\left|x\right|$: absolute value of $x$

Referenced by:

(1.16.32), item (ii), item (ii)

Permalink:

http://dlmf.nist.gov/1.16.E30

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.15):

Previously this equation was completely different:

$${𝐷}_{𝜶}={\mathrm{i}}^{-\left|𝜶\right|}{𝐷}^{𝜶}={\left(\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_1}\right)}^{{\alpha }_1}\mathrm{\cdots }{\left(\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_n}\right)}^{{\alpha }_n}.$$

The change was made to clarify the meaning of (1.16.32).

See also:

Annotations for §1.16(vii), §1.16 and Ch.1

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14: 19.31 Probability Distributions

§19.31 Probability Distributions

► $R_G(x,y,z)$ and $R_F(x,y,z)$ occur as the expectation values, relative to a normal probability distribution in ${ℝ}^2$ or ${ℝ}^3$, of the square root or reciprocal square root of a quadratic form. …§19.16(iii) shows that for $n=3$ the incomplete cases of $R_F$ and $R_G$ occur when $\mu =-1/2$ and $\mu =1/2$, respectively, while their complete cases occur when $n=2$. …

15: Bibliography O

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Tables of Fourier Transforms and Fourier Transforms of Distributions. Springer-Verlag, Berlin.

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External Links:

ISBN 3-540-50630-6; 0-387-50630-6, MathReview, ZentralBlatt

Referenced by:

§1.14(viii), §10.22(vi), §10.43(vi), §11.10(x), §11.7(v), §11.9(iv), §12.12, §13.10(iv), §13.23(ii), §18.17(ix), §6.14(iii), §7.14(iii).

Encodings:

BibTeX

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On the distribution of spacings between zeros of the zeta function. Math. Comp. 48 (177), pp. 273–308.

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External Links:

ISSN 0025-5718, FullText, Author’s Website, MathReview (S. L. Segal), ZentralBlatt

Referenced by:

§25.18(ii).

Encodings:

BibTeX

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Complete elliptic integrals resulting from infinite integrals of Bessel functions. J. Res. Nat. Bur. Standards Sect. B 78B (3), pp. 113–135.

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External Links:

ISSN 0160-1741, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§10.22(vi), §10.43(vi).

Encodings:

BibTeX

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Complete elliptic integrals resulting from infinite integrals of Bessel functions. II. J. Res. Nat. Bur. Standards Sect. B 79B (3-4), pp. 137–170.

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External Links:

ISSN 0160-1741, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

§10.22(vi), §10.43(vi).

Encodings:

BibTeX

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Distribution of the partition function modulo $m$ . Ann. of Math. (2) 151 (1), pp. 293–307.

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External Links:

ISSN 0003-486X, FullText, MathReview (Bruce C. Berndt), ZentralBlatt

Referenced by:

§27.14(v).

Encodings:

BibTeX

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16: Bibliography H

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Sur la distribution des zéros de la fonction $\zeta (s)$ et ses conséquences arithmétiques. Bull. Soc. Math. France 24, pp. 199–220 (French).

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Notes:

Reprinted in Oeuvres de Jacques Hadamard. Tomes I, pp. 189–210, Éditions du Centre National de la Recherche Scientifique, Paris, 1968.

External Links:

ISSN 0037-9484, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§25.10(i), §27.2(i).

Encodings:

BibTeX

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On the distribution of the zeros of generalized Airy functions. Math. Comp. 29 (131), pp. 863–877.

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External Links:

FullText, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§9.13(i).

Encodings:

BibTeX

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Algorithm 55: Complete elliptic integral of the first kind. Comm. ACM 4 (4), pp. 180.

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Notes:

Includes Algol program, maximum accuracy 8D. For certification see same journal 6 (1966), pp. 166–167

External Links:

ISSN 0001-0782, Document

Referenced by:

§19.39(ii).

Encodings:

BibTeX

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Algorithm 56: Complete elliptic integral of the second kind. Comm. ACM 4 (4), pp. 180–181.

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Notes:

Includes Algol program, maximum accuracy 8D. For certification see same journal 9 (1966), p. 12

External Links:

ISSN 0001-0782, Document

Referenced by:

§19.39(ii).

Encodings:

BibTeX

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Algorithm 395: Student’s t-distribution. Comm. ACM 13 (10), pp. 617–619.

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Notes:

Includes Algol program, maximum accuracy 11S. For Remarks see ACM Trans. Math. Software 5(1979) and 7(1981)

External Links:

ISSN 0001-0782, Document, Remark 1, Remark 2

Referenced by:

§8.28(iv).

Encodings:

BibTeX

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17: Bibliography F

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Notes:

The “Freely Distributable LIBM” package provides implementations of standard elementary functions plus a few higher functions, e.g. gamma. Double precision, maximum accuracy 20S. Developed by Sun Microsystems.

External Links:

GAMS (package FDLIBM)

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

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A Table of the Complete Elliptic Integral of the First Kind for Complex Values of the Modulus. Part I. Technical report Technical Report ARL 69-0172, Aerospace Research Laboratories, Office of Aerospace Research, Wright-Patterson Air Force Base, Ohio.

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Notes:

Table erratum: Math. Comp. v. 36 (1981), no. 153, p. 318. Part II of this report, with the same date but numbered ARL 69-0173, again puts $k=R\exp (i\theta )$ but with $R$ as parameter and $\theta$ as variable instead of the other way around. Part III, dated May 1970 and numbered ARL 70-0081, contains tables of auxiliary functions to help interpolation.

External Links:

MathReview, ZentralBlatt

Referenced by:

§19.36(ii), §19.37(ii).

Encodings:

BibTeX

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Mémoire sur la diffraction de la lumière. Mém. de l’Académie des Sciences, pp. 247–382.

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Notes:

Œvres Complètes d’Augustin Fresnel, Paris, 1866

Referenced by:

Figure 7.3.4, Figure 7.3.4.

Encodings:

BibTeX

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18: Bibliography M

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The exponential integral distribution. Statist. Probab. Lett. 5 (3), pp. 209–211.

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External Links:

ISSN 0167-7152, Document, MathReview Entry, ZentralBlatt

Referenced by:

§8.19(xi).

Encodings:

BibTeX

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Algorithm 149: Complete elliptic integral. Comm. ACM 5 (12), pp. 605.

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Notes:

Includes Algol program, maximum accuracy 10D. For remark see ACM Trans. Math. Software 4 (1978), p. 95

External Links:

ISSN 0001-0782, Document

Referenced by:

§19.39(ii).

Encodings:

BibTeX

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Calculation of the complete elliptic integrals with complex modulus. Numer. Math. 29 (2), pp. 233–236.

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Notes:

An Algol procedure for calculating the complete elliptic integrals of the first and the second kind with complex modulus $k$ is included.

External Links:

ISSN 0029-599X, Document, MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§19.36(ii), §19.39(ii).

Encodings:

BibTeX

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19: 18.2 General Orthogonal Polynomials

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§18.2(vi) Zeros

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§18.2(viii) Uniqueness of Orthogonality Measure and Completeness

… ►A system $\{p_n(x)\}$ of OP’s satisfying (18.2.1) and (18.2.5) is complete if each $f(x)$ in the Hilbert space $L_w^2((a,b))$ can be approximated in Hilbert norm by finite sums ${\sum }_n{\lambda }_n{p}_n(x)$. …A system of OP’s with unique orthogonality measure is always complete, see Shohat and Tamarkin (1970, Theorem 2.14). In particular, a system of OP’s on a bounded interval is always complete. …