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21: 6.7 Integral Representations

§6.7 Integral Representations

… ► ►

§6.7(ii) Sine and Cosine Integrals

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§6.7(iii) Auxiliary Functions

… ►For collections of integral representations see Bierens de Haan (1939, pp. 56–59, 72–73, 82–84, 121, 133–136, 155, 179–181, 223, 225–227, 230, 259–260, 374, 377, 397–398, 408, 416, 424, 431, 438–439, 442–444, 488, 496–500, 567–571, 585, 602, 638, 675–677), Corrington (1961), Erdélyi et al. (1954a, vol. 1, pp. 267–270), Geller and Ng (1969), Nielsen (1906b), Oberhettinger (1974, pp. 244–246), Oberhettinger and Badii (1973, pp. 364–371), and Watrasiewicz (1967).

22: 7.7 Integral Representations

§7.7 Integral Representations

►

§7.7(i) Error Functions and Dawson’s Integral

… ►

§7.7(ii) Auxiliary Functions

… ►

Mellin–Barnes Integrals

… ►For other integral representations see Erdélyi et al. (1954a, vol. 1, pp. 265–267, 270), Ng and Geller (1969), Oberhettinger (1974, pp. 246–248), and Oberhettinger and Badii (1973, pp. 371–377).

23: 5.14 Multidimensional Integrals

§5.14 Multidimensional Integrals

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5.14.7

\frac{1}{(2\pi)^{n}}\int_{[-\pi,\pi]^{n}}\prod_{1\leq j<k\leq n}|e^{i\theta_{j% }}-e^{i\theta_{k}}|^{2b}\,\mathrm{d}\theta_{1}\cdots\,\mathrm{d}\theta_{n}=% \frac{\Gamma\left(1+bn\right)}{(\Gamma\left(1+b\right))^{n}},

\Re b>-1/n

ⓘ

Symbols:: $\Gamma\left(\NVar{z}\right)$ : gamma function, $\pi$ : the ratio of the circumference of a circle to its diameter, $[\NVar{a},\NVar{b}]$ : closed interval, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part, $j$ : nonnegative integer, $n$ : nonnegative integer, $k$ : nonnegative integer and $b$ : real or complex variable
Referenced by:: §5.14, §5.20
Permalink:: http://dlmf.nist.gov/5.14.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §5.14, §5.14 and Ch.5

24: 15.17 Mathematical Applications

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§15.17(iii) Group Representations

…

25: 16.24 Physical Applications

… ►The

\mathit{3j}

symbols, or Clebsch–Gordan coefficients, play an important role in the decomposition of reducible representations of the rotation group into irreducible representations. …

26: 13.12 Products

… ►For integral representations, integrals, and series containing products of

M\left(a,b,z\right)

and

U\left(a,b,z\right)

see Erdélyi et al. (1953a, §6.15.3).

27: Morris Newman

… ►Newman wrote the book Matrix Representations of Groups, published by the National Bureau of Standards in 1968, and the book Integral Matrices, published by Academic Press in 1972, which became a classic. …