Gaussian elimination
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11: 26.9 Integer Partitions: Restricted Number and Part Size
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26.9.4
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►is the Gaussian polynomial (or -binomial coefficient); see also §§17.2(i)–17.2(ii).
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26.9.5
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26.9.6
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26.9.7
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12: 3.5 Quadrature
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►As in Simpson’s rule, by combining the rule for with that for , the first error term in (3.5.9) can be eliminated.
With the Romberg scheme successive terms , in (3.5.9) are eliminated, according to the formula
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►For effective testing of Gaussian quadrature rules see Gautschi (1983).
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►Oscillatory integral transforms are treated in Wong (1982) by a method based on Gaussian quadrature.
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13: 8.24 Physical Applications
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►The function appears in: discussions of power-law relaxation times in complex physical systems (Sornette (1998)); logarithmic oscillations in relaxation times for proteins (Metzler et al. (1999)); Gaussian orbitals and exponential (Slater) orbitals in quantum chemistry (Shavitt (1963), Shavitt and Karplus (1965)); population biology and ecological systems (Camacho et al. (2002)).
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14: 20.13 Physical Applications
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►Then the nonperiodic Gaussian
…Thus the classical theta functions are “periodized”, or “anti-periodized”, Gaussians; see Bellman (1961, pp. 18, 19).
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15: 1.13 Differential Equations
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Elimination of First Derivative by Change of Dependent Variable
… ►Elimination of First Derivative by Change of Independent Variable
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1.13.17
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16: 35.1 Special Notation
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►The main functions treated in this chapter are the multivariate gamma and beta functions, respectively and , and the special functions of matrix argument: Bessel (of the first kind) and (of the second kind) ; confluent hypergeometric (of the first kind) or and (of the second kind) ; Gaussian hypergeometric or ; generalized hypergeometric or .
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►Related notations for the Bessel functions are (Faraut and Korányi (1994, pp. 320–329)), (Terras (1988, pp. 49–64)), and (Faraut and Korányi (1994, pp. 357–358)).
complex variables. | |
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17: 7.21 Physical Applications
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►Voigt functions , , can be regarded as the convolution of a Gaussian and a Lorentzian, and appear when the analysis of light (or particulate) absorption (or emission) involves thermal motion effects.
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18: 26.16 Multiset Permutations
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►The
-multinomial coefficient is defined in terms of Gaussian polynomials (§26.9(ii)) by
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26.16.1
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