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1: 36.9 Integral Identities

§36.9 Integral Identities

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36.9.9 $${\left|{\mathrm{\Psi }}^{(\mathrm{E})}\left(x,y,z\right)\right|}^2=\frac{8{\pi }^2}{3^{2/3}}{\int }_0^{\mathrm{\infty }}{\int }_0^{2\pi }\mathrm{\Re }\left(\mathrm{Ai}\left(\frac{1}{3^{1/3}}\left(x+\mathrm{i}y+2zu\exp \left(\mathrm{i}\theta \right)+3u^2\exp \left(-2\mathrm{i}\theta \right)\right)\right)\mathrm{Bi}\left(\frac{1}{3^{1/3}}\left(x-\mathrm{i}y+2zu\exp \left(-\mathrm{i}\theta \right)+3u^2\exp \left(2\mathrm{i}\theta \right)\right)\right)\right)udud\theta .$$

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Symbols:

$\mathrm{Ai}\left(z\right)$: Airy function, $\mathrm{Bi}\left(z\right)$: Airy function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , ${\mathrm{\Psi }}^{(\mathrm{E})}\left(𝐱\right)$: elliptic umbilic canonical integral function, $\exp z$: exponential function, $\mathrm{i}$: imaginary unit, $\int$: integral, $\mathrm{\Re }$: real part, $y$: real parameter, $z$: real parameter and $x$: real parameter

Keywords:

elliptic umbilic canonical integral, integralidentity

Permalink:

http://dlmf.nist.gov/36.9.E9

Encodings:

pMML, png, TeX

See also:

Annotations for §36.9 and Ch.36

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2: Bibliography O

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A class of integral identities with matrix argument. Duke Math. J. 26 (2), pp. 207–213.

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External Links:

Document, MathReview (R. Bellman), ZentralBlatt

Referenced by:

§35.3(ii).

Encodings:

BibTeX

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3: Bibliography W

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An algorithmic proof theory for hypergeometric (ordinary and “$q$”) multisum/integral identities. Invent. Math. 108, pp. 575–633.

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External Links:

ISSN 0020-9910, Document, MathReview (David R. Masson), ZentralBlatt

Referenced by:

§16.4(iii).

Encodings:

BibTeX

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Rational function certification of multisum/integral/“$q$” identities. Bull. Amer. Math. Soc. (N.S.) 27 (1), pp. 148–153.

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External Links:

ISSN 0273-0979, Pdf, Document, MathReview (Robert A. Gustafson), ZentralBlatt

Referenced by:

§16.4(iii).

Encodings:

BibTeX

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4: 19.1 Special Notation

… ► $R_F(x,y,z)$, $R_G(x,y,z)$, and $R_J(x,y,z,p)$ are the symmetric (in $x$, $y$, and $z$) integrals of the first, second, and third kinds; they are complete if exactly one of $x$, $y$, and $z$ is identically 0. …

5: 1.4 Calculus of One Variable

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1.4.32

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Symbols:

$dx$: differential of $x$ , $\equiv$: equals by definition, $\int$: integral and $\left|x\right|$: absolute value of $x$

Permalink:

http://dlmf.nist.gov/1.4.E32

Encodings:

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See also:

Annotations for §1.4(v), §1.4(v), §1.4 and Ch.1

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6: 24.7 Integral Representations

§24.7 Integral Representations

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§24.7(i) Bernoulli and Euler Numbers

►The identities in this subsection hold for $n=1,2,\mathrm{\dots }$. … ►

§24.7(ii) Bernoulli and Euler Polynomials

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Mellin–Barnes Integral

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7: 35.10 Methods of Computation

… ►Other methods include numerical quadrature applied to double and multiple integral representations. See Yan (1992) for the ${}_1{}^{}F_1^{}$ and ${}_2{}^{}F_1^{}$ functions of matrix argument in the case $m=2$, and Bingham et al. (1992) for Monte Carlo simulation on $𝐎(m)$ applied to a generalization of the integral (35.5.8). ►Koev and Edelman (2006) utilizes combinatorial identities for the zonal polynomials to develop computational algorithms for approximating the series expansion (35.8.1). …

8: 18.40 Methods of Computation

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18.40.4 $$\underset{N\to \mathrm{\infty }}{lim}{F}_N(z)=F(z)\equiv \frac{1}{{\mu }_0}{\int }_a^b\frac{w(x)dx}{z-x},$$ $z\in ℂ\backslash [a,b]$,

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Symbols:

$[a,b]$: closed interval, $ℂ$: complex plane, $dx$: differential of $x$ , $\in$: element of, $\equiv$: equals by definition, $\int$: integral, $N$: positive integer, $w(x)$: weight function, $z$: complex variable and $x$: real variable

Proved:

Ismail (2009, p. 36)(proved)

Permalink:

http://dlmf.nist.gov/18.40.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §18.40(ii), §18.40(ii), §18.40 and Ch.18

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9: 36.10 Differential Equations

§36.10 Differential Equations

… ► $K=1$, fold: (36.10.6) is an identity. ► $K=2$, cusp: … ► $K=3$, swallowtail: … ►In terms of the normal forms (36.2.2) and (36.2.3), the ${\mathrm{\Psi }}^{(\mathrm{U})}\left(𝐱\right)$ satisfy the following operator equations …

10: 25.12 Polylogarithms

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25.12.11 $${\mathrm{Li}}_s\left(z\right)\equiv \frac{z}{\mathrm{\Gamma }\left(s\right)}{\int }_0^{\mathrm{\infty }}\frac{x^{s-1}}{{\mathrm{e}}^x-z}dx,$$

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $dx$: differential of $x$ , $\equiv$: equals by definition, $\mathrm{e}$: base of natural logarithm, $\int$: integral, ${\mathrm{Li}}_s\left(z\right)$: polylogarithm, $x$: real variable, $s$: complex variable and $z$: complex variable

Keywords:

improper integral, integral representation

Source:

Lewin (1981, (7.188), p. 236)

Referenced by:

(25.12.16), §25.12(ii)

Permalink:

http://dlmf.nist.gov/25.12.E11

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(ii), §25.12(ii), §25.12 and Ch.25

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