pseudo-spectral representations
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11: 13.27 Mathematical Applications
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βΊConfluent hypergeometric functions are connected with representations of the group of third-order triangular matrices.
…Vilenkin (1968, Chapter 8) constructs irreducible representations of this group, in which the diagonal matrices correspond to operators of multiplication by an exponential function.
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12: 14.32 Methods of Computation
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βΊIn particular, for small or moderate values of the parameters and the power-series expansions of the various hypergeometric function representations given in §§14.3(i)–14.3(iii), 14.19(ii), and 14.20(i) can be selected in such a way that convergence is stable, and reasonably rapid, especially when the argument of the functions is real.
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13: 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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14: 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
…15: 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).16: 24.7 Integral Representations
§24.7 Integral Representations
βΊ§24.7(i) Bernoulli and Euler Numbers
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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).17: 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
…18: 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.
19: 23.11 Integral Representations
§23.11 Integral Representations
…20: 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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