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uniform asymptotic approximations for large parameters

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1: 28.26 Asymptotic Approximations for Large q
§28.26 Asymptotic Approximations for Large q
§28.26(ii) Uniform Approximations
2: 28.8 Asymptotic Expansions for Large q
§28.8 Asymptotic Expansions for Large q
Barrett’s Expansions
Dunster’s Approximations
3: 13.8 Asymptotic Approximations for Large Parameters
§13.8(ii) Large b and z , Fixed a and b / z
4: 8.12 Uniform Asymptotic Expansions for Large Parameter
§8.12 Uniform Asymptotic Expansions for Large Parameter
Inverse Function
5: 13.20 Uniform Asymptotic Approximations for Large μ
§13.20(i) Large μ , Fixed κ
6: 13.21 Uniform Asymptotic Approximations for Large κ
§13.21 Uniform Asymptotic Approximations for Large κ
7: 13.22 Zeros
Asymptotic approximations to the zeros when the parameters κ and/or μ are large can be found by reversion of the uniform approximations provided in §§13.20 and 13.21. …
8: 12.10 Uniform Asymptotic Expansions for Large Parameter
§12.10 Uniform Asymptotic Expansions for Large Parameter
§12.10(vi) Modifications of Expansions in Elementary Functions
Modified Expansions
9: 14.26 Uniform Asymptotic Expansions
§14.26 Uniform Asymptotic Expansions
The uniform asymptotic approximations given in §14.15 for P ν - μ ( x ) and Q ν μ ( x ) for 1 < x < 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). See also Frenzen (1990), Gil et al. (2000), Shivakumar and Wong (1988), Ursell (1984), and Wong (1989) for uniform asymptotic approximations obtained from integral representations.
10: Bibliography O
  • A. B. Olde Daalhuis (2003a) Uniform asymptotic expansions for hypergeometric functions with large parameters. I. Analysis and Applications (Singapore) 1 (1), pp. 111–120.
  • A. B. Olde Daalhuis (2003b) Uniform asymptotic expansions for hypergeometric functions with large parameters. II. Analysis and Applications (Singapore) 1 (1), pp. 121–128.
  • A. B. Olde Daalhuis (2010) Uniform asymptotic expansions for hypergeometric functions with large parameters. III. Analysis and Applications (Singapore) 8 (2), pp. 199–210.
  • F. W. J. Olver (1959) Uniform asymptotic expansions for Weber parabolic cylinder functions of large orders. J. Res. Nat. Bur. Standards Sect. B 63B, pp. 131–169.
  • F. W. J. Olver (1980b) 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.