approximations
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31: 13.31 Approximations
§13.31 Approximations
… ►§13.31(ii) Padé Approximations
… ►§13.31(iii) Rational Approximations
►In Luke (1977a) the following rational approximation is given, together with its rate of convergence. …32: 28.26 Asymptotic Approximations for Large
§28.26 Asymptotic Approximations for Large
… ►§28.26(ii) Uniform Approximations
… ►For asymptotic approximations for see also Naylor (1984, 1987, 1989).33: 13.22 Zeros
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►Asymptotic approximations to the zeros when the parameters and/or are large can be found by reversion of the uniform approximations provided in §§13.20 and 13.21.
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34: 24.18 Physical Applications
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►Euler polynomials also appear in statistical physics as well as in semi-classical approximations to quantum probability distributions (Ballentine and McRae (1998)).
35: 29.20 Methods of Computation
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►Initial approximations to the eigenvalues can be found, for example, from the asymptotic expansions supplied in §29.7(i).
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►A third method is to approximate eigenvalues and Fourier coefficients of Lamé functions by eigenvalues and eigenvectors of finite matrices using the methods of §§3.2(vi) and 3.8(iv).
…The approximations converge geometrically (§3.8(i)) to the eigenvalues and coefficients of Lamé functions as .
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►A fourth method is by asymptotic approximations by zeros of orthogonal polynomials of increasing degree.
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36: 33.23 Methods of Computation
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►Inside the turning points, that is, when , there can be a loss of precision by a factor of approximately
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§33.23(vii) WKBJ Approximations
►WKBJ approximations (§2.7(iii)) for are presented in Hull and Breit (1959) and Seaton and Peach (1962: in Eq. …Seaton (1984) estimates the accuracies of these approximations. ►Hull and Breit (1959) and Barnett (1981b) give WKBJ approximations for and in the region inside the turning point: .37: 33.21 Asymptotic Approximations for Large
§33.21 Asymptotic Approximations for Large
►§33.21(i) Limiting Forms
… ►§33.21(ii) Asymptotic Expansions
…38: 2.1 Definitions and Elementary Properties
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►means that for each , the difference between and the th partial sum on the right-hand side is as in .
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►Some asymptotic approximations are expressed in terms of two or more Poincaré asymptotic expansions.
…For an example see (2.8.15).
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§2.1(iv) Uniform Asymptotic Expansions
… ►§2.1(v) Generalized Asymptotic Expansions
…39: 2.2 Transcendental Equations
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►Higher approximations are obtainable by successive resubstitutions.
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2.2.6
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►For other examples see de Bruijn (1961, Chapter 2).