uniform asymptotic expansions for large parameter
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11: 28.26 Asymptotic Approximations for Large
§28.26 Asymptotic Approximations for Large
►§28.26(i) Goldstein’s Expansions
… ►The asymptotic expansions of and in the same circumstances are also given by the right-hand sides of (28.26.4) and (28.26.5), respectively. … ►§28.26(ii) Uniform Approximations
… ►For asymptotic approximations for see also Naylor (1984, 1987, 1989).12: 13.20 Uniform Asymptotic Approximations for Large
§13.20 Uniform Asymptotic Approximations for Large
►§13.20(i) Large , Fixed
… ► … ►§13.20(v) Large , Other Expansions
… ►13: Bibliography W
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Uniform asymptotic expansion of via a difference equation.
Numer. Math. 91 (1), pp. 147–193.
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Estimates for the error term in a uniform asymptotic expansion of the Jacobi polynomials.
Anal. Appl. (Singap.) 1 (2), pp. 213–241.
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Uniform asymptotic expansion of the Jacobi polynomials in a complex domain.
Proc. Roy. Soc. London Ser. A 460, pp. 2569–2586.
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An asymptotic expansion of with large variable and parameters.
Math. Comp. 27 (122), pp. 429–436.
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On uniform asymptotic expansion of definite integrals.
J. Approximation Theory 7 (1), pp. 76–86.
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14: 14.26 Uniform Asymptotic Expansions
§14.26 Uniform Asymptotic Expansions
►The 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). ►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.15: 10.41 Asymptotic Expansions for Large Order
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10.41.4
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16: 13.8 Asymptotic Approximations for Large Parameters
§13.8 Asymptotic Approximations for Large Parameters
… ►§13.8(ii) Large and , Fixed and
… ►§13.8(iii) Large
… ► … ►§13.8(iv) Large and
…17: Bibliography T
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Uniform asymptotic expansions of confluent hypergeometric functions.
J. Inst. Math. Appl. 22 (2), pp. 215–223.
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Laplace type integrals: Transformation to standard form and uniform asymptotic expansions.
Quart. Appl. Math. 43 (1), pp. 103–123.
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Uniform asymptotic expansions of integrals: A selection of problems.
J. Comput. Appl. Math. 65 (1-3), pp. 395–417.
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Numerical algorithms for uniform Airy-type asymptotic expansions.
Numer. Algorithms 15 (2), pp. 207–225.
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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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18: 10.72 Mathematical Applications
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►Bessel functions and modified Bessel functions are often used as approximants in the construction of uniform asymptotic approximations and expansions for solutions of linear second-order differential equations containing a parameter.
…where is a real or complex variable and is a large real or complex parameter.
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►In regions in which (10.72.1) has a simple turning point , that is, and are analytic (or with weaker conditions if is a real variable) and is a simple zero of , asymptotic expansions of the solutions for large
can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order (§9.6(i)).
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►If has a double zero , or more generally is a zero of order , , then uniform asymptotic approximations (but not expansions) can be constructed in terms of Bessel functions, or modified Bessel functions, of order .
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►These asymptotic expansions are uniform with respect to , including cut neighborhoods of , and again the region of uniformity often includes cut neighborhoods of other singularities of the differential equation.
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19: 10.69 Uniform Asymptotic Expansions for Large Order
§10.69 Uniform Asymptotic Expansions for Large Order
… ►All fractional powers take their principal values. ►All four expansions also enjoy the same kind of double asymptotic property described in §10.41(iv). … ►20: 2.8 Differential Equations with a Parameter
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►in which is a real or complex parameter, and asymptotic solutions are needed for large
that are uniform with respect to in a point set in or .
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►For other examples of uniform asymptotic approximations and expansions of special functions in terms of Bessel functions or modified Bessel functions of fixed order see §§13.8(iii), 13.21(i), 13.21(iv), 14.15(i), 14.15(iii), 14.20(vii), 15.12(iii), 18.15(i), 18.15(iv), 18.24, 33.20(iv).
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►However, in all cases with and or , only uniform asymptotic approximations are available, not uniform asymptotic expansions.
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►For further examples of uniform asymptotic approximations in terms of parabolic cylinder functions see §§13.20(iii), 13.20(iv), 14.15(v), 15.12(iii), 18.24.
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►For examples of uniform asymptotic approximations in terms of Whittaker functions with fixed second parameter see §18.15(i) and §28.8(iv).
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