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11: 15.17 Mathematical Applications

… ►First, as spherical functions on noncompact Riemannian symmetric spaces of rank one, but also as associated spherical functions, intertwining functions, matrix elements of SL$(2,ℝ)$, and spherical functions on certain nonsymmetric Gelfand pairs. …

12: 35.2 Laplace Transform

… ►

35.2.1 $$g(𝐙)={\int }_{𝛀}\mathrm{etr}\left(-𝐙𝐗\right)f(𝐗)d𝐗,$$

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Defines:

$g$: Laplace transformation of $f$ (locally)

Symbols:

$\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, $𝐗$: real symmetric $m\times m$ matrix, $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices, $𝐙$: complex symmetric $m\times m$ matrix and $f(𝐗)$: complex-valued function of matrix argument

Keywords:

Laplace transform

Referenced by:

§35.2

Permalink:

http://dlmf.nist.gov/35.2.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §35.2, §35.2 and Ch.35

►where the integration variable $𝐗$ ranges over the space $𝛀$. …

13: 35.4 Partitions and Zonal Polynomials

… ►

35.4.3 $$Z_{\kappa }\left(𝐓\right)=Z_{\kappa }\left(𝐈\right){\left|𝐓\right|}^{k_m}\underset{𝐎(m)}{\int }\prod _{j=1}^{m-1}{|{(𝐇𝐓{𝐇}^{-1})}_j|}^{k_j-k_{j+1}}\mathrm{d}𝐇,$$ $𝐓\in 𝓢$.

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Symbols:

$\in$: element of, $\int$: integral, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $𝐈$: $m\times m$ identity matrix, $𝓢$: space of real symmetric $m\times m$ matrices, $𝐓$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $|{(𝐗)}_j|$: $j$th principal minor of $𝐗$ , $𝐎(m)$: space of $m\times m$ orthogonal matrices, $𝐇$: orthogonal $m\times m$ matrix, $\mathrm{d}𝐇$: normalized Haar measure on $𝐎(m)$ , $j$: nonnegative integer, $k$: nonnegative integer, $m$: positive integer and $\kappa$: vector of nonnegative integers

Referenced by:

§35.4(i)

Permalink:

http://dlmf.nist.gov/35.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §35.4(i), §35.4 and Ch.35

… ►

35.4.5 $$Z_{\kappa }\left(𝐇𝐓{𝐇}^{-1}\right)=Z_{\kappa }\left(𝐓\right),$$ $𝐇\in 𝐎(m)$.

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Symbols:

$\in$: element of, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $𝐓$: real symmetric $m\times m$ matrix, $𝐎(m)$: space of $m\times m$ orthogonal matrices, $𝐇$: orthogonal $m\times m$ matrix, $m$: positive integer and $\kappa$: vector of nonnegative integers

Permalink:

http://dlmf.nist.gov/35.4.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

… ►

35.4.7 $${\int }_{𝐎(m)}{Z}_{\kappa }\left(𝐒𝐇𝐓{𝐇}^{-1}\right)\mathrm{d}𝐇=\frac{Z_{\kappa }\left(𝐒\right)Z_{\kappa }\left(𝐓\right)}{Z_{\kappa }\left(𝐈\right)}.$$

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Symbols:

$\int$: integral, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $𝐈$: $m\times m$ identity matrix, $𝐒$: real symmetric $m\times m$ matrix, $𝐓$: real symmetric $m\times m$ matrix, $𝐎(m)$: space of $m\times m$ orthogonal matrices, $𝐇$: orthogonal $m\times m$ matrix, $\mathrm{d}𝐇$: normalized Haar measure on $𝐎(m)$ , $m$: positive integer and $\kappa$: vector of nonnegative integers

Referenced by:

§35.9

Permalink:

http://dlmf.nist.gov/35.4.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

… ►

35.4.8 $${\int }_{𝛀}\mathrm{etr}\left(-𝐓𝐗\right){\left|𝐗\right|}^{a-\frac{1}{2}(m+1)}{Z}_{\kappa }\left(𝐗\right)d𝐗={\mathrm{\Gamma }}_m\left(a+\kappa \right){\left|𝐓\right|}^{-a}{Z}_{\kappa }\left({𝐓}^{-1}\right),$$

ⓘ

Symbols:

$\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $a$: complex variable, $𝐓$: real symmetric $m\times m$ matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices, $m$: positive integer and $\kappa$: vector of nonnegative integers

Permalink:

http://dlmf.nist.gov/35.4.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

…

14: 35.3 Multivariate Gamma and Beta Functions

… ►

35.3.1 $${\mathrm{\Gamma }}_m\left(a\right)={\int }_{𝛀}\mathrm{etr}\left(-𝐗\right){\left|𝐗\right|}^{a-\frac{1}{2}(m+1)}d𝐗,$$ $\mathrm{\Re }\left(a\right)>\frac{1}{2}(m-1)$.

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Symbols:

$\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.3.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §35.3(i), §35.3 and Ch.35

►

35.3.2 $${\mathrm{\Gamma }}_m\left(s_1,\mathrm{\dots },s_m\right)={\int }_{𝛀}\mathrm{etr}\left(-𝐗\right){\left|𝐗\right|}^{s_m-\frac{1}{2}(m+1)}\prod _{j=1}^{m-1}{|{(𝐗)}_j|}^{s_j-s_{j+1}}d𝐗,$$ $s_j\in ℂ$, $\mathrm{\Re }\left(s_j\right)>\frac{1}{2}(j-1)$, $j=1,\mathrm{\dots },m$.

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Symbols:

$ℂ$: complex plane, $\in$: element of, $\mathrm{etr}\left(𝐀\right)$: exponential of trace, $\int$: integral, ${\mathrm{\Gamma }}_m\left(a\right)$: multivariate gamma function, $\mathrm{\Re }$: real part, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $|{(𝐗)}_j|$: $j$th principal minor of $𝐗$ , $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices, $j$: nonnegative integer and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.3.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §35.3(i), §35.3 and Ch.35

… ►

35.3.8 $${\mathrm{B}}_m(a,b)={\int }_{𝛀}{\left|𝐗\right|}^{a-\frac{1}{2}(m+1)}{\left|𝐈+𝐗\right|}^{-(a+b)}d𝐗,$$ $\mathrm{\Re }\left(a\right),\mathrm{\Re }\left(b\right)>\frac{1}{2}(m-1)$.

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Symbols:

$\int$: integral, ${\mathrm{B}}_m(a,b)$: multivariate beta function, $\mathrm{\Re }$: real part, $a$: complex variable, $𝐈$: $m\times m$ identity matrix, $𝐗$: real symmetric $m\times m$ matrix, $\left|𝐗\right|$: determinant of $𝐗$ , $d$: differential of matrix, $𝛀$: space of positive-definite real symmetric $m\times m$ matrices, $b$: complex variable and $m$: positive integer

Permalink:

http://dlmf.nist.gov/35.3.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §35.3(ii), §35.3 and Ch.35

15: Bibliography O

… ►

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

… ►

Triorthogonal systems in spaces of constant curvature in which the equation ${\mathrm{\Delta }}_{2}u+\lambda u=0$ allows a complete separation of variables. Mat. Sbornik N.S. 27(69) (3), pp. 379–426 (Russian).

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Notes:

In Russian.

External Links:

MathReview (H. Busemann), ZentralBlatt

Referenced by:

§31.17(i).

Encodings:

BibTeX

… ►

Solution of Equations in Euclidean and Banach Spaces. Pure and Applied Mathematics, Vol. 9, Academic Press, New York-London.

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Notes:

Third edition of Solution of equations and systems of equations

External Links:

MathReview (J. Burlak), ZentralBlatt

Referenced by:

§3.3(v), §3.8(iii).

Encodings:

BibTeX

…

16: Bibliography T

… ►

Harmonic Analysis on Symmetric Spaces and Applications. II. Springer-Verlag, Berlin.

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External Links:

ISBN 3-540-96663-3, MathReview (Stephen Gelbart), ZentralBlatt

Referenced by:

§35.1, §35.5(ii), Notations K.

Encodings:

BibTeX

… ►

Special functions in phase space: Mathieu functions. J. Phys. A 31 (31), pp. 6725–6739.

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External Links:

ISSN 0305-4470, Document, MathReview (Bing Hong Wang), ZentralBlatt

Referenced by:

8th item.

Encodings:

BibTeX

►

Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 (1), pp. 151–174.

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External Links:

ISSN 0010-3616, FullText, MathReview (Estelle L. Basor), ZentralBlatt

Referenced by:

§32.14.

Encodings:

BibTeX

…

17: Bibliography Y

… ►

Mathematical Apparatus of the Theory of Angular Momentum. Israel Program for Scientific Translations for National Science Foundation and the National Aeronautics and Space Administration, Jerusalem.

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Notes:

Translated from the Russian by A. Sen and R. N. Sen

External Links:

MathReview (R. F. Stora), ZentralBlatt

Referenced by:

§34.11.

Encodings:

BibTeX

18: 3.4 Differentiation

… ►

§3.4(i) Equally-Spaced Nodes

… ►For formulas for derivatives with equally-spaced real nodes and based on Sinc approximations (§3.3(vi)), see Stenger (1993, §3.5). …

19: 36.4 Bifurcation Sets

… ►This is the codimension-one surface in $𝐱$ space where critical points coalesce, satisfying (36.4.1) and … ►This is the codimension-one surface in $𝐱$ space where critical points coalesce, satisfying (36.4.2) and …

20: 36.7 Zeros

… ►The zeros are lines in $𝐱=(x,y,z)$ space where $\mathrm{ph}{\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$ is undetermined. … ►The zeros of these functions are curves in $𝐱=(x,y,z)$ space; see Nye (2007) for ${\mathrm{\Phi }}_3$ and Nye (2006) for ${\mathrm{\Phi }}^{(\mathrm{H})}$.