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21: 3.3 Interpolation
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βΊwhere is a simple closed contour in described in the positive rotational sense and enclosing the points .
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βΊand are the Lagrangian interpolation coefficients defined by
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βΊwhere is given by (3.3.3), and is a simple closed contour in described in the positive rotational sense and enclosing .
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βΊBy using this approximation to as a new point, , and evaluating , we find that , with 9 correct digits.
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βΊThen by using in Newton’s interpolation formula, evaluating and recomputing , another application of Newton’s rule with starting value gives the approximation , with 8 correct digits.
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22: 16.24 Physical Applications
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βΊ
§16.24(iii) , , and Symbols
… βΊThey can be expressed as functions with unit argument. …These are balanced functions with unit argument. Lastly, special cases of the symbols are functions with unit argument. …23: 9.4 Maclaurin Series
24: Bibliography S
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Orthogonal polynomials arising in the numerical evaluation of inverse Laplace transforms.
Math. Tables Aids Comput. 9 (52), pp. 164–177.
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Evaluation of associated Legendre functions off the cut and parabolic cylinder functions.
Electron. Trans. Numer. Anal. 9, pp. 137–146.
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Parabolic Cylinder Functions and their Application in Symmetric Two-centre Shell Model.
In Proceedings of the Conference on Mathematical Analysis and its
Applications (Inst. Engrs., Mysore, 1977),
Matscience Rep., Vol. 91, Aarhus, pp. P81–P89.
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A new Fortran program for the - angular momentum coefficient.
Comput. Phys. Comm. 56 (2), pp. 231–248.
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An Introduction to Basic Fourier Series.
Developments in Mathematics, Vol. 9, Kluwer Academic Publishers, Dordrecht.
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25: 19.36 Methods of Computation
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βΊIf (19.36.1) is used instead of its first five terms, then the factor in Carlson (1995, (2.2)) is changed to .
βΊFor both and the factor in Carlson (1995, (2.18)) is changed to when the following polynomial of degree 7 (the same for both) is used instead of its first seven terms:
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βΊAll cases of , , , and are computed by essentially the same procedure (after transforming Cauchy principal values by means of (19.20.14) and (19.2.20)).
…Because of cancellations in (19.26.21) it is advisable to compute from and by (19.21.10) or else to use §19.36(ii).
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βΊAccurate values of for near 0 can be obtained from by (19.2.6) and (19.25.13).
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26: 16.26 Approximations
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βΊFor discussions of the approximation of generalized hypergeometric functions and the Meijer -function in terms of polynomials, rational functions, and Chebyshev polynomials see Luke (1975, §§5.12 - 5.13) and Luke (1977b, Chapters 1 and 9).
27: 16.7 Relations to Other Functions
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βΊFor , , symbols see Chapter 34.
Further representations of special functions in terms of functions are given in Luke (1969a, §§6.2–6.3), and an extensive list of functions with rational numbers as parameters is given in Krupnikov and Kölbig (1997).
28: 34.13 Methods of Computation
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βΊMethods of computation for and symbols include recursion relations, see Schulten and Gordon (1975a), Luscombe and Luban (1998), and Edmonds (1974, pp. 42–45, 48–51, 97–99); summation of single-sum expressions for these symbols, see Varshalovich et al. (1988, §§8.2.6, 9.2.1) and Fang and Shriner (1992); evaluation of the generalized hypergeometric functions of unit argument that represent these symbols, see Srinivasa Rao and Venkatesh (1978) and Srinivasa Rao (1981).
βΊFor symbols, methods include evaluation of the single-sum series (34.6.2), see Fang and Shriner (1992); evaluation of triple-sum series, see Varshalovich et al. (1988, §10.2.1) and Srinivasa Rao et al. (1989).
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