asymptotic approximations for large parameters
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11: 34.8 Approximations for Large Parameters
§34.8 Approximations for Large Parameters
βΊFor large values of the parameters in the , , and symbols, different asymptotic forms are obtained depending on which parameters are large. … βΊSemiclassical (WKBJ) approximations in terms of trigonometric or exponential functions are given in Varshalovich et al. (1988, §§8.9, 9.9, 10.7). Uniform approximations in terms of Airy functions for the and symbols are given in Schulten and Gordon (1975b). For approximations for the , , and symbols with error bounds see Flude (1998), Chen et al. (1999), and Watson (1999): these references also cite earlier work.12: 25.11 Hurwitz Zeta Function
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§25.11(xii) -Asymptotic Behavior
… βΊAs in the sector , with and fixed, we have the asymptotic expansion … βΊSimilarly, as in the sector , βΊ
25.11.44
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25.11.45
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13: 12.10 Uniform Asymptotic Expansions for Large Parameter
§12.10 Uniform Asymptotic Expansions for Large Parameter
… βΊ§12.10(vi) Modifications of Expansions in Elementary Functions
… βΊ … βΊModified Expansions
… βΊ14: 14.26 Uniform Asymptotic Expansions
§14.26 Uniform Asymptotic Expansions
βΊThe uniform asymptotic approximations given in §14.15 for and for are extended to domains in the complex plane in the following references: §§14.15(i) and 14.15(ii), Dunster (2003b); §14.15(iii), Olver (1997b, Chapter 12); §14.15(iv), Boyd and Dunster (1986). For an extension of §14.15(iv) to complex argument and imaginary parameters, see Dunster (1990b). 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.15: 12.11 Zeros
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§12.11(ii) Asymptotic Expansions of Large Zeros
… βΊWhen the zeros are asymptotically given by and , where is a large positive integer and … βΊ§12.11(iii) Asymptotic Expansions for Large Parameter
βΊFor large negative values of the real zeros of , , , and can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). … βΊ
12.11.4
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16: 15.12 Asymptotic Approximations
§15.12 Asymptotic Approximations
βΊ§15.12(i) Large Variable
… βΊ§15.12(ii) Large
… βΊAs , … βΊFor other extensions, see Wagner (1986), Temme (2003) and Temme (2015, Chapters 12 and 28).17: Bibliography O
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Uniform asymptotic expansions for hypergeometric functions with large parameters. I.
Analysis and Applications (Singapore) 1 (1), pp. 111–120.
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Uniform asymptotic expansions for hypergeometric functions with large parameters. II.
Analysis and Applications (Singapore) 1 (1), pp. 121–128.
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Uniform asymptotic expansions for hypergeometric functions with large parameters. III.
Analysis and Applications (Singapore) 8 (2), pp. 199–210.
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Asymptotic approximations and error bounds.
SIAM Rev. 22 (2), pp. 188–203.
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Whittaker functions with both parameters large: Uniform approximations in terms of parabolic cylinder functions.
Proc. Roy. Soc. Edinburgh Sect. A 86 (3-4), pp. 213–234.
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18: Bibliography T
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Laguerre polynomials: Asymptotics for large degree.
Technical report
Technical Report AM-R8610, CWI, Amsterdam, The Netherlands.
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On the computation of the incomplete gamma functions for large values of the parameters.
In Algorithms for approximation (Shrivenham, 1985),
Inst. Math. Appl. Conf. Ser. New Ser., Vol. 10, pp. 479–489.
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Computational aspects of incomplete gamma functions with large complex parameters.
In Approximation and Computation. A Festschrift in Honor
of Walter Gautschi, R. V. M. Zahar (Ed.),
International Series of Numerical Mathematics, Vol. 119, pp. 551–562.
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Large parameter cases of the Gauss hypergeometric function.
J. Comput. Appl. Math. 153 (1-2), pp. 441–462.
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Asymptotic expansions of Kummer hypergeometric functions for large values of the parameters.
Integral Transforms Spec. Funct. 33 (1), pp. 16–31.
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19: 18.15 Asymptotic Approximations
§18.15 Asymptotic Approximations
βΊ§18.15(i) Jacobi
… βΊ§18.15(ii) Ultraspherical
… βΊ§18.15(iii) Legendre
… βΊ§18.15(iv) Laguerre
…20: 18.26 Wilson Class: Continued
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