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11: 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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12: 19.2 Definitions
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βΊThe integral for is well defined if , and the Cauchy principal value (§1.4(v)) of is taken if vanishes at an interior point of the integration path.
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βΊ
§19.2(iv) A Related Function:
… βΊFormulas involving that are customarily different for circular cases, ordinary hyperbolic cases, and (hyperbolic) Cauchy principal values, are united in a single formula by using . … βΊWhen and are positive, is an inverse circular function if and an inverse hyperbolic function (or logarithm) if : …For the special cases of and see (19.6.15). …13: 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. …14: 18.8 Differential Equations
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βΊ
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15: 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).
16: 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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17: 34.9 Graphical Method
§34.9 Graphical Method
… βΊFor specific examples of the graphical method of representing sums involving the , and symbols, see Varshalovich et al. (1988, Chapters 11, 12) and Lehman and O’Connell (1973, §3.3).18: 34.10 Zeros
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βΊSuch zeros are called nontrivial zeros.
βΊFor further information, including examples of nontrivial zeros and extensions to symbols, see Srinivasa Rao and Rajeswari (1993, pp. 133–215, 294–295, 299–310).
19: 34.7 Basic Properties: Symbol
§34.7 Basic Properties: Symbol
… βΊ§34.7(ii) Symmetry
… βΊ§34.7(iv) Orthogonality
… βΊ§34.7(vi) Sums
… βΊIt constitutes an addition theorem for the symbol. …20: 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)).
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βΊThe incomplete integrals and can be computed by successive transformations in which two of the three variables converge quadratically to a common value and the integrals reduce to , accompanied by two quadratically convergent series in the case of ; compare Carlson (1965, §§5,6).
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βΊ
can be evaluated by using (19.25.5).
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