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21: 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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1.16.32 $$P(𝐃)=\sum _{𝜶}{c}_{𝜶}{𝐃}^{\alpha }=\sum _{𝜶}{c}_{𝜶}{\left(\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_1}\right)}^{{\alpha }_1}\mathrm{\dots }{\left(\frac{1}{\mathrm{i}}\frac{\partial }{\partial x_n}\right)}^{{\alpha }_n}.$$

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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, $𝐃$: vector differential operator, $P$: polynomial of several variables and $𝐷f$: distributional derivative

Referenced by:

(1.16.30), (1.16.33), (1.16.34), (1.16.36), (1.16.37), item (iii), item (iii)

Permalink:

http://dlmf.nist.gov/1.16.E32

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.15):

Previously this equation was written

$$P(𝐷)=\sum _{𝜶}{c}_{𝜶}{𝐷}_{𝜶}$$

where ${𝐷}_{𝜶}$ was defined in (1.16.30). The change was made to clarify the meaning of this equation.

See also:

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

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1.16.33 $$ℱ(P(𝐃)\varphi )(𝐱)=P(-𝐱)ℱ\varphi (𝐱),$$

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Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\int$: integral, $ℝ$: real line, $n$: nonnegative integer, $\varphi (x_1,x_2,\mathrm{\dots },x_n)$: test function, $ℱ(\varphi )$: Fourier transform of a test function, $𝐃$: vector differential operator, $P$: polynomial of several variables and $𝐷f$: distributional derivative

Referenced by:

item (iv), §1.16(vii), item (iv)

Permalink:

http://dlmf.nist.gov/1.16.E33

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

Previously this equation was written as

$$\frac{1}{{(2\pi )}^{n/2}}{\int }_{{ℝ}^n}(P(𝐷)\varphi )(𝐭){\mathrm{e}}^{\mathrm{i}𝐱\cdot 𝐭}d𝐭=P(-𝐱)F(𝐱).$$

It has been rewritten using the notation of (1.16.29) and (1.16.32).

See also:

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

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1.16.36 $$⟨ℱ\left(P(𝐃)u\right),\varphi ⟩=⟨P_{-}ℱ\left(u\right),\varphi ⟩=⟨ℱ\left(u\right),P_{-}\varphi ⟩,$$

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Symbols:

$⟨\mathrm{\Lambda },\varphi ⟩$: action of distribution on test function, $ℱ\left(u\right)$: Fourier transform of a tempered distribution, $\varphi (x_1,x_2,\mathrm{\dots },x_n)$: test function, $𝐃$: vector differential operator, $P$: polynomial of several variables and $𝐷f$: distributional derivative

Referenced by:

item (iv), §1.16(vii), §1.16(viii), item (iv)

Permalink:

http://dlmf.nist.gov/1.16.E36

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

Previously this equation was written as

$$ℱ(P(𝐷)u)=P(-𝐱)ℱ(u).$$

It has been rewritten using the notation of (1.16.32) and (1.16.35).

See also:

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

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1.16.37 $$⟨ℱ\left(Pu\right),\varphi ⟩=⟨P(𝐃)ℱ\left(u\right),\varphi ⟩,$$

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Symbols:

$⟨\mathrm{\Lambda },\varphi ⟩$: action of distribution on test function, $ℱ\left(u\right)$: Fourier transform of a tempered distribution, $\varphi (x_1,x_2,\mathrm{\dots },x_n)$: test function, $𝐃$: vector differential operator, $P$: polynomial of several variables and $𝐷f$: distributional derivative

Referenced by:

item (iv), §1.16(vii), item (iv)

Permalink:

http://dlmf.nist.gov/1.16.E37

Encodings:

pMML, png, TeX

Notational Change (effective with 1.0.15):

Previously this equation was written as

$$ℱ(Pu)=P(𝐷)ℱ(u).$$

It has been rewritten using the notation of (1.16.32) and (1.16.35).

See also:

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

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22: Bibliography D

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Introduction to Vector Analysis. 5th edition, Allyn and Bacon Inc., Boston, MA.

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Notes:

Seventh edition printed by WCB/McGraw-Hill, 1995.

External Links:

ISBN 0-697-06356-9; 0-697-16099-8, MathReview (R. H. Bowman), ZentralBlatt

Referenced by:

§1.5(ii).

Encodings:

BibTeX

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Vector coupling coefficients as products of prime factors. Comput. Phys. Comm. 4 (2), pp. 268–274.

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Notes:

Single-precision Fortran

External Links:

Document, Code

Referenced by:

4th item.

Encodings:

BibTeX

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23: Bibliography H

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Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach. 2nd edition, Prentice Hall Inc., Upper Saddle River, NJ.

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External Links:

ISBN 0-13-041408-5; 978-0-13-041408-3, MathReview (W. P. Ziemer), ZentralBlatt

Referenced by:

§1.6(i).

Encodings:

BibTeX

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24: 35.5 Bessel Functions of Matrix Argument

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35.5.2 $$A_{\nu }\left(𝐓\right)=A_{\nu }\left(𝟎\right)\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k}{k!}\sum _{|\kappa |=k}\frac{1}{{\left[\nu +\frac{1}{2}(m+1)\right]}_{\kappa }}{Z}_{\kappa }\left(𝐓\right),$$ $\nu \in ℂ$, $𝐓\in 𝓢$.

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Symbols:

$A_{\nu }\left(𝐓\right)$: Bessel function of matrix argument (first kind), $ℂ$: complex plane, $\in$: element of, $!$: factorial (as in $n!$), ${\left[a\right]}_{\kappa }$: partitional shifted factorial, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $𝓢$: space of real symmetric $m\times m$ matrices, $𝐓$: real symmetric $m\times m$ matrix, $k$: nonnegative integer, $m$: positive integer, $𝟎$: $m\times m$ matrix of zeros and $\kappa$: vector of nonnegative integers

Permalink:

http://dlmf.nist.gov/35.5.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §35.5(i), §35.5 and Ch.35

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25: Bibliography K

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Nonsymmetric Askey-Wilson polynomials as vector-valued polynomials. Appl. Anal. 90 (3-4), pp. 731–746.

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External Links:

ISSN 0003-6811, Link, MathReview (Masatoshi Iida), ZentralBlatt

Referenced by:

§18.38(iii), §18.38(iii).

Encodings:

BibTeX

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26: Bibliography L

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VOIGTL – A fast subroutine for Voigt function evaluation on vector processors. Comput. Phys. Comm. 75 (1-2), pp. 135–142.

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Notes:

Single-precision Fortran, minimum accuracy 13D.

External Links:

Document, Code, ZentralBlatt

Referenced by:

3rd item.

Encodings:

BibTeX

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27: 1.11 Zeros of Polynomials

… ►where the column vector $h_k^{(m)}$ consists of the first $k$ members of the sequence $a_m,a_{m-1},a_{m-2},\mathrm{\dots }$ with $a_j=0$ if or $j>n$. …

28: 35.6 Confluent Hypergeometric Functions of Matrix Argument

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35.6.1 $${}_1{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a}{b};𝐓)=\sum _{k=0}^{\mathrm{\infty }}\frac{1}{k!}\sum _{|\kappa |=k}\frac{{\left[a\right]}_{\kappa }}{{\left[b\right]}_{\kappa }}{Z}_{\kappa }\left(𝐓\right).$$

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $!$: factorial (as in $n!$), ${\left[a\right]}_{\kappa }$: partitional shifted factorial, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $a$: complex variable, $𝐓$: real symmetric $m\times m$ matrix, $b$: complex variable, $k$: nonnegative integer and $\kappa$: vector of nonnegative integers

Permalink:

http://dlmf.nist.gov/35.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §35.6(i), §35.6 and Ch.35

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29: 35.7 Gaussian Hypergeometric Function of Matrix Argument

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35.7.1 $${}_2{}^{}F_1^{}(\genfrac{}{}{0pt}{}{a,b}{c};𝐓)=\sum _{k=0}^{\mathrm{\infty }}\frac{1}{k!}\sum _{|\kappa |=k}\frac{{\left[a\right]}_{\kappa }{\left[b\right]}_{\kappa }}{{\left[c\right]}_{\kappa }}{Z}_{\kappa }\left(𝐓\right),$$ $-c+\frac{1}{2}(j+1)\notin \mathbb{N}$, $1\le j\le m$; .

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Symbols:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument, $!$: factorial (as in $n!$), $\notin$: not an element of, $\mathbb{N}$: set of all positive integers, ${\left[a\right]}_{\kappa }$: partitional shifted factorial, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $a$: complex variable, $𝐓$: real symmetric $m\times m$ matrix, $b$: complex variable, $\Vert 𝐓\Vert$: spectral norm of $𝐓$ , $c$: complex variable, $j$: nonnegative integer, $k$: nonnegative integer, $m$: positive integer and $\kappa$: vector of nonnegative integers

Permalink:

http://dlmf.nist.gov/35.7.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §35.7(i), §35.7 and Ch.35

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30: 35.8 Generalized Hypergeometric Functions of Matrix Argument

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35.8.1 $${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)=\sum _{k=0}^{\mathrm{\infty }}\frac{1}{k!}\sum _{|\kappa |=k}\frac{{\left[a_1\right]}_{\kappa }\mathrm{\cdots }{\left[a_p\right]}_{\kappa }}{{\left[b_1\right]}_{\kappa }\mathrm{\cdots }{\left[b_q\right]}_{\kappa }}{Z}_{\kappa }\left(𝐓\right).$$

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Defines:

${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}(\genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q};𝐓)$: generalized hypergeometric function of matrix argument

Symbols:

$!$: factorial (as in $n!$), ${\left[a\right]}_{\kappa }$: partitional shifted factorial, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $a$: complex variable, $𝐓$: real symmetric $m\times m$ matrix, $b$: complex variable, $k$: nonnegative integer, $p$: nonnegative integer, $q$: nonnegative integer and $\kappa$: vector of nonnegative integers

Referenced by:

§35.10, §35.10, §35.8(i), §35.8(i), §35.8(i), §35.8(i)

Permalink:

http://dlmf.nist.gov/35.8.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §35.8(i), §35.8 and Ch.35

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