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21: 5.20 Physical Applications

… ►Suppose the potential energy of a gas of $n$ point charges with positions $x_1,x_2,\mathrm{\dots },x_n$ and free to move on the infinite line , is given by … ►

5.20.3 $${\psi }_n(\beta )={\int }_{{ℝ}^n}{\mathrm{e}}^{-\beta W}dx={(2\pi )}^{n/2}{\beta }^{-(n/2)-(\beta n(n-1)/4)}{\left(\mathrm{\Gamma }\left(1+\frac{1}{2}\beta \right)\right)}^{-n}\prod _{j=1}^n\mathrm{\Gamma }\left(1+\frac{1}{2}j\beta \right).$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\int$: integral, $ℝ$: real line, $j$: nonnegative integer, $n$: nonnegative integer, $x$: real variable, $W$: potential energy, ${\psi }_n(\beta )$: partition function and $\beta =1/(kT)$

Permalink:

http://dlmf.nist.gov/5.20.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §5.20, §5.20 and Ch.5

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22: 8.6 Integral Representations

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§8.6(i) Integrals Along the Real Line

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23: 11.5 Integral Representations

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§11.5(i) Integrals Along the Real Line

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24: 12.5 Integral Representations

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§12.5(i) Integrals Along the Real Line

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25: 18.32 OP’s with Respect to Freud Weights

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18.32.2 $$w(x)={\left|x\right|}^{\alpha }\exp \left(-Q(x)\right),$$ $x\in ℝ$, $\alpha >-1$,

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Symbols:

$\in$: element of, $\exp z$: exponential function, $ℝ$: real line, $w(x)$: weight function and $x$: real variable

Referenced by:

§18.32, Erratum (V1.2.0) Section 18.32

Permalink:

http://dlmf.nist.gov/18.32.E2

Encodings:

pMML, png, TeX

Addition (effective with 1.2.0):

This equation was added.

See also:

Annotations for §18.32 and Ch.18

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26: 19.31 Probability Distributions

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19.31.2 $${\int }_{{ℝ}^n}{({𝐱}^{\mathrm{T}}𝐀𝐱)}^{\mu }\exp \left(-{𝐱}^{\mathrm{T}}𝐁𝐱\right)d{x}_1\mathrm{\cdots }d{x}_n=\frac{{\pi }^{n/2}\mathrm{\Gamma }\left(\mu +\frac{1}{2}n\right)}{\sqrt{det𝐁}\mathrm{\Gamma }\left(\frac{1}{2}n\right)}{R}_{\mu }(\frac{1}{2},\mathrm{\dots },\frac{1}{2};{\lambda }_1,\mathrm{\dots },{\lambda }_n),$$ $\mu >-\frac{1}{2}n$.

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Symbols:

$R_{-a}(b_1,\mathrm{\dots },b_n;z_1,\mathrm{\dots },z_n)$ or $R_{-a}(𝐛;𝐳)$: multivariate hypergeometric function, $\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $det$: determinant, $dx$: differential of $x$ , $\exp z$: exponential function, $\int$: integral, $ℝ$: real line, ${𝐀}^{\mathrm{T}}$: transpose of matrix, $n$: nonnegative integer and ${\lambda }_n$: eigenvalues

Referenced by:

§19.31

Permalink:

http://dlmf.nist.gov/19.31.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §19.31 and Ch.19

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27: Bibliography N

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Dislocation lines in the hyperbolic umbilic diffraction catastrophe. Proc. Roy. Soc. Lond. Ser. A 462, pp. 2299–2313.

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External Links:

ISSN 1364-5021, Document, MathReview, ZentralBlatt

Referenced by:

§36.7(iv).

Encodings:

BibTeX

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Dislocation lines in the swallowtail diffraction catastrophe. Proc. Roy. Soc. Lond. Ser. A 463, pp. 343–355.

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External Links:

ISSN 1364-5021, Document, ZentralBlatt

Referenced by:

§36.7(iv).

Encodings:

BibTeX

28: 19.6 Special Cases

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29: 22.19 Physical Applications

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22.19.6 $$x(t)=a\mathrm{cn}(t\sqrt{1+2\eta },k).$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $k$: modulus, $x(t)$: coordinate, $t$: time, $a$: displacement and $\eta$: parameter

Referenced by:

Erratum (V1.0.9) for Equation (22.19.6), Erratum (V1.0.9) for Equations (22.19.6), (22.19.7), (22.19.8), (22.19.9)

Permalink:

http://dlmf.nist.gov/22.19.E6

Encodings:

pMML, png, TeX

Errata (effective with 1.0.9):

Originally the term $\sqrt{1+2\eta }$ was given incorrectly as $\sqrt{1+\eta }$ in this equation and in the line above. Additionally, for improved clarity, the modulus $k$ has been defined in the line above. Previously $1/\sqrt{2+{\eta }^{-1}}$ was given explicitly in the equation.

Reported 2014-04-28 by Svante Janson

See also:

Annotations for §22.19(ii), §22.19(ii), §22.19 and Ch.22

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22.19.7 $$x(t)=a\mathrm{sn}(t\sqrt{1-\eta },k).$$

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Symbols:

$\mathrm{sn}(z,k)$: Jacobian elliptic function, $k$: modulus, $x(t)$: coordinate, $t$: time, $a$: displacement and $\eta$: parameter

Referenced by:

Erratum (V1.0.9) for Equations (22.19.6), (22.19.7), (22.19.8), (22.19.9)

Permalink:

http://dlmf.nist.gov/22.19.E7

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.9):

For added clarity, the modulus $k$ has been defined in the line above. Previously $1/\sqrt{{\eta }^{-1}-1}$ was given explicitly in the equation.

See also:

Annotations for §22.19(ii), §22.19(ii), §22.19 and Ch.22

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22.19.9 $$x(t)=a\mathrm{cn}(t\sqrt{2\eta -1},k),$$

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Symbols:

$\mathrm{cn}(z,k)$: Jacobian elliptic function, $k$: modulus, $x(t)$: coordinate, $t$: time, $a$: displacement and $\eta$: parameter

Referenced by:

Erratum (V1.0.9) for Equations (22.19.6), (22.19.7), (22.19.8), (22.19.9)

Permalink:

http://dlmf.nist.gov/22.19.E9

Encodings:

pMML, png, TeX

Clarification (effective with 1.0.9):

For added clarity, the modulus $k$ has been defined in the line below. Previously $1/\sqrt{2-{\eta }^{-1}}$ was given explicitly in the equation.

See also:

Annotations for §22.19(ii), §22.19(ii), §22.19 and Ch.22

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30: 10.32 Integral Representations

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§10.32(i) Integrals along the Real Line

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10.32.11 $$K_{\nu }\left(xz\right)=\frac{\mathrm{\Gamma }\left(\nu +\frac{1}{2}\right){(2z)}^{\nu }}{{\pi }^{\frac{1}{2}}{x}^{\nu }}{\int }_0^{\mathrm{\infty }}\frac{\cos \left(xt\right)dt}{{(t^2+z^2)}^{\nu +\frac{1}{2}}},$$ $\mathrm{\Re }\nu >-\frac{1}{2}$, $x>0$, .

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\pi$: the ratio of the circumference of a circle to its diameter, $\cos z$: cosine function, $dx$: differential of $x$ , $\int$: integral, $K_{\nu }\left(z\right)$: modified Bessel function of the second kind, $\mathrm{ph}$: phase, $\mathrm{\Re }$: real part, $x$: real variable, $z$: complex variable and $\nu$: complex parameter

A&S Ref:

9.6.25 (corrected)

Permalink:

http://dlmf.nist.gov/10.32.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §10.32(i), §10.32(i), §10.32 and Ch.10

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