Euler-product representation
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11: 34.6 Definition: Symbol
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βΊThe symbol may be defined either in terms of symbols or equivalently in terms of symbols:
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34.6.1
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βΊThe symbol may also be written as a finite triple sum equivalent to a terminating generalized hypergeometric series of three variables with unit arguments.
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12: 1.17 Integral and Series Representations of the Dirac Delta
§1.17 Integral and Series Representations of the Dirac Delta
… βΊ§1.17(ii) Integral Representations
… βΊThen comparison of (1.17.2) and (1.17.9) yields the formal integral representation … βΊSine and Cosine Functions
… βΊ§1.17(iii) Series Representations
…13: 14.25 Integral Representations
§14.25 Integral Representations
… βΊFor corresponding contour integrals, with less restrictions on and , see Olver (1997b, pp. 174–179), and for further integral representations see Magnus et al. (1966, §4.6.1).14: 24.7 Integral Representations
§24.7 Integral Representations
βΊ§24.7(i) Bernoulli and Euler Numbers
… βΊ
24.7.5
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§24.7(ii) Bernoulli and Euler Polynomials
… βΊFor further integral representations see Prudnikov et al. (1986a, §§2.3–2.6) and Gradshteyn and Ryzhik (2000, Chapters 3 and 4).15: 25.5 Integral Representations
§25.5 Integral Representations
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25.5.5
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βΊFor similar representations involving other theta functions see Erdélyi et al. (1954a, p. 339).
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25.5.19
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§25.5(iii) Contour Integrals
…16: 14.26 Uniform Asymptotic Expansions
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βΊSee also Frenzen (1990), Gil et al. (2000), Shivakumar and Wong (1988), Ursell (1984), and Wong (1989) for uniform asymptotic approximations obtained from integral representations.
17: 23.11 Integral Representations
§23.11 Integral Representations
…18: 35.10 Methods of Computation
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βΊOther methods include numerical quadrature applied to double and multiple integral representations.
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19: Bibliography V
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Representation of Lie Groups and Special Functions. Volume 2: Class I Representations, Special Functions, and Integral Transforms.
Mathematics and its Applications (Soviet Series), Vol. 74, Kluwer Academic Publishers Group, Dordrecht.
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Special Functions and the Theory of Group Representations.
American Mathematical Society, Providence, RI.
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Representation of an odd number as a sum of three primes (Russian).
Dokl. Akad. Nauk SSSR 15, pp. 169–172 (Russian).
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Integral representations for products of Lamé functions by use of fundamental solutions.
SIAM J. Math. Anal. 15 (3), pp. 559–569.
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Fourier series representation of Ferrers function
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20: Donald St. P. Richards
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βΊHe is editor of the book Hypergeometric Functions on Domains of Positivity, Jack Polynomials, and Applications, published by the American Mathematical Society in 1992, and coeditor of Representation Theory and Harmonic Analysis: A Conference in Honor of R. A. Kunze (with T.
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