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11: 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, integral identity

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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12: Possible Errors in DLMF

… ►Errors in the printed Handbook may already have been corrected in the online version; please consult Errata. …

13: Bibliography L

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Lie algebraic approaches to classical partition identities. Adv. in Math. 29 (1), pp. 15–59.

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External Links:

ISSN 0001-8708, Document, MathReview (S. I. Gel’fand), ZentralBlatt

Referenced by:

§17.16.

Encodings:

BibTeX

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A Lie theoretic interpretation and proof of the Rogers-Ramanujan identities. Adv. in Math. 45 (1), pp. 21–72.

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External Links:

ISSN 0001-8708, Document, MathReview (George E. Andrews), ZentralBlatt

Referenced by:

§17.16.

Encodings:

BibTeX

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On Bessel functions of equal order and argument. Rend. Sem. Mat. Univ. Politec. Torino 50 (2), pp. 209–216 (1993).

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External Links:

ISSN 0373-1243, MathReview (N. Hayek Calil), ZentralBlatt

Referenced by:

§10.14.

Encodings:

BibTeX

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Numerical Evaluation of Special Functions. In Mathematics of Computation 1943–1993: A Half-Century of Computational Mathematics (Vancouver, BC, 1993), Proc. Sympos. Appl. Math., Vol. 48, pp. 79–125.

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Notes:

An updated version exists online.

External Links:

FullText, MathReview (John P. Coleman), ZentralBlatt

Referenced by:

Ch.3.

Encodings:

BibTeX

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14: 9.18 Tables

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Yakovleva (1969) tabulates Fock’s functions $U(x)\equiv \sqrt{\pi }\mathrm{Bi}\left(x\right)$, $U^{\prime }(x)=\sqrt{\pi }{\mathrm{Bi}}^{\prime }\left(x\right)$, $V(x)\equiv \sqrt{\pi }\mathrm{Ai}\left(x\right)$, $V^{\prime }(x)=\sqrt{\pi }{\mathrm{Ai}}^{\prime }\left(x\right)$ for $x=-9(.001)9$. Precision is 7S.

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Miller (1946) tabulates $a_k$, ${\mathrm{Ai}}^{\prime }\left(a_k\right)$, $a_k^{\prime }$, $\mathrm{Ai}\left(a_k^{\prime }\right)$, $k=1(1)50$; $b_k$, ${\mathrm{Bi}}^{\prime }\left(b_k\right)$, $b_k^{\prime }$, $\mathrm{Bi}\left(b_k^{\prime }\right)$, $k=1(1)20$. Precision is 8D. Entries for $k=1(1)20$ are reproduced in Abramowitz and Stegun (1964, Chapter 10).

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Sherry (1959) tabulates $a_k$, ${\mathrm{Ai}}^{\prime }\left(a_k\right)$, $a_k^{\prime }$, $\mathrm{Ai}\left(a_k^{\prime }\right)$, $k=1(1)50$; 20S.

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National Bureau of Standards (1958) tabulates $A_0(x)\equiv \pi \mathrm{Hi}\left(-x\right)$ and $-A_0^{\prime }(x)\equiv \pi {\mathrm{Hi}}^{\prime }\left(-x\right)$ for $x=0(.01)1(.02)5(.05)11$ and $1/x=0.01(.01)0.1$; ${\int }_0^x{A}_0(t)dt$ for $x=0.5,1(1)11$. Precision is 8D.

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Nosova and Tumarkin (1965) tabulates $e_0(x)\equiv \pi \mathrm{Hi}\left(-x\right)$, $e_0^{\prime }(x)=-\pi {\mathrm{Hi}}^{\prime }\left(-x\right)$, ${\tilde{e}}_0(-x)\equiv -\pi \mathrm{Gi}\left(x\right)$, ${\tilde{e}}_0^{\prime }(-x)=\pi {\mathrm{Gi}}^{\prime }\left(x\right)$ for $x=-1(.01)10$; 7D. Also included are the real and imaginary parts of $e_0(z)$ and $\mathrm{i}{e}_0^{\prime }(z)$, where $z=\mathrm{i}y$ and $y=0(.01)9$; 6-7D.

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15: 22.9 Cyclic Identities

§22.9 Cyclic Identities

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§22.9(ii) Typical Identities of Rank 2

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§22.9(iii) Typical Identities of Rank 3

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16: 24.5 Recurrence Relations

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§24.5(ii) Other Identities

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§24.5(iii) Inversion Formulas

►In each of (24.5.9) and (24.5.10) the first identity implies the second one and vice-versa. …

17: 15.17 Mathematical Applications

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§15.17(iv) Combinatorics

►In combinatorics, hypergeometric identities classify single sums of products of binomial coefficients. …

18: How to Cite

… ►The direct correspondence between the reference numbers in the printed Handbook and the permalinks used online in the DLMF enables readers of either version to cite specific items and their readers to easily look them up again — in either version! ►The following table outlines the correspondence between reference numbers as they appear in the Handbook, and the URL’s that find the same item online. …

19: 27.15 Chinese Remainder Theorem

… ►The Chinese remainder theorem states that a system of congruences $x\equiv a_1(mod{m}_1),\mathrm{\dots },x\equiv a_k(mod{m}_k)$, always has a solution if the moduli are relatively prime in pairs; the solution is unique (mod $m$), where $m$ is the product of the moduli. …

20: Bibliography U

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Mathematics of Computation Unpublished Mathematical Tables Collection.

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Notes:

Archives of American Mathematics, Center for American History, The University of Texas at Austin. Covers 1944–1994. An online guide is arranged chronologically by date of issuance within Mathematics of Computation

External Links:

Guide

Referenced by:

§27.21.

Encodings:

BibTeX

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Collisions and umbilic catastrophes. The hyperbolic umbilic canonical diffraction integral. Molecular Phys. 50 (6), pp. 1215–1230.

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External Links:

ISSN 0026-8976, Document, MathReview (D. R. J. Chillingworth)

Referenced by:

§36.14(iii).

Encodings:

BibTeX