asymptotic approximations and expansions for large %7Cr%7C
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31: 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.
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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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►In regions in which the function has a simple pole at and is analytic at (the case in §10.72(i)), asymptotic expansions of the solutions of (10.72.1) for large
can be constructed in terms of Bessel functions and modified Bessel functions of order , where is the limiting value of as .
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►Then for large
asymptotic approximations of the solutions can be constructed in terms of Bessel functions, or modified Bessel functions, of variable order (in fact the order depends on and ).
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32: 12.9 Asymptotic Expansions for Large Variable
§12.9 Asymptotic Expansions for Large Variable
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12.9.1
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12.9.2
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12.9.3
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§12.9(ii) Bounds and Re-Expansions for the Remainder Terms
…33: 34.8 Approximations for Large Parameters
§34.8 Approximations for Large Parameters
►For large values of the parameters in the , , and symbols, different asymptotic forms are obtained depending on which parameters are large. …34: 11.11 Asymptotic Expansions of Anger–Weber Functions
§11.11 Asymptotic Expansions of Anger–Weber Functions
►§11.11(i) Large , Fixed
… ► ►§11.11(ii) Large , Fixed
… ►Lastly, corresponding asymptotic approximations and expansions for and , with or , follow from (11.10.15) and (11.10.16) and the corresponding asymptotic expansions for the Bessel functions and ; see §10.19(ii). …35: Bibliography F
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Uniform asymptotic expansions for hypergeometric functions with large parameters IV.
Anal. Appl. (Singap.) 12 (6), pp. 667–710.
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Asymptotic expansions of a class of hypergeometric polynomials with respect to the order. II.
J. Math. Anal. Appl. 7 (3), pp. 440–451.
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Uniform asymptotic expansions of certain classes of Meijer -functions for a large parameter.
SIAM J. Math. Anal. 4 (3), pp. 482–507.
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Uniform asymptotic expansions of a class of Meijer -functions for a large parameter.
SIAM J. Math. Anal. 14 (6), pp. 1204–1253.
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On the asymptotic expansion of Mellin transforms.
SIAM J. Math. Anal. 18 (1), pp. 273–282.
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36: Bibliography L
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A note on the uniform asymptotic expansion of integrals with coalescing endpoint and saddle points.
J. Phys. A 19 (3), pp. 329–335.
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Error bounds for asymptotic expansions of Laplace convolutions.
SIAM J. Math. Anal. 25 (6), pp. 1537–1553.
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Uniform asymptotic expansions of symmetric elliptic integrals.
Constr. Approx. 17 (4), pp. 535–559.
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Asymptotic expansions of the Whittaker functions for large order parameter.
Methods Appl. Anal. 6 (2), pp. 249–256.
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Uniform asymptotic expansions at a caustic.
Comm. Pure Appl. Math. 19, pp. 215–250.
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37: 10.19 Asymptotic Expansions for Large Order
§10.19 Asymptotic Expansions for Large Order
►§10.19(i) Asymptotic Forms
… ►§10.19(ii) Debye’s Expansions
… ►§10.19(iii) Transition Region
… ►See also §10.20(i).38: 15.12 Asymptotic Approximations
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