asymptotic%20expansions%20for%20large%20variable
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1: Bibliography B
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Asymptotic expansions of the modified Bessel function of the third kind of imaginary order.
SIAM J. Appl. Math. 15, pp. 1315–1323.
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Chapter 13 of Ramanujan’s second notebook: Integrals and asymptotic expansions.
Expo. Math. 2 (4), pp. 289–347.
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Uniform asymptotic expansion of Charlier polynomials.
Methods Appl. Anal. 1 (3), pp. 294–313.
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Problem of two Coulomb centres at large intercentre separation: Asymptotic expansions from analytical solutions of the Heun equation.
J. Phys. A 30 (2), pp. 559–571.
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On the asymptotic expansion of some integrals.
Arch. Math. (Basel) 42 (3), pp. 253–259.
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2: 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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A further method for the evaluation of zeros of Bessel functions and some new asymptotic expansions for zeros of functions of large order.
Proc. Cambridge Philos. Soc. 47, pp. 699–712.
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Some new asymptotic expansions for Bessel functions of large orders.
Proc. Cambridge Philos. Soc. 48 (3), pp. 414–427.
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3: 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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The third Appell function for one large variable.
J. Approx. Theory 165, pp. 60–69.
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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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4: 9.9 Zeros
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►If is regarded as a continuous variable, then
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§9.9(iv) Asymptotic Expansions
►For large … ►
9.9.18
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►For error bounds for the asymptotic expansions of , , , and see Pittaluga and Sacripante (1991), and a conjecture given in Fabijonas and Olver (1999).
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5: 30.9 Asymptotic Approximations and Expansions
§30.9 Asymptotic Approximations and Expansions
►§30.9(i) Prolate Spheroidal Wave Functions
… ►For uniform asymptotic expansions in terms of Airy or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1986). … ►For uniform asymptotic expansions in terms of elementary, Airy, or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1992, 1995). … ►6: 28.8 Asymptotic Expansions for Large
§28.8 Asymptotic Expansions for Large
… ►§28.8(ii) Sips’ Expansions
… ►§28.8(iii) Goldstein’s Expansions
… ►Barrett’s Expansions
… ►7: 9.7 Asymptotic Expansions
§9.7 Asymptotic Expansions
… ►§9.7(iii) Error Bounds for Real Variables
… ►§9.7(iv) Error Bounds for Complex Variables
… ►§9.7(v) Exponentially-Improved Expansions
… ►8: 11.6 Asymptotic Expansions
§11.6 Asymptotic Expansions
►§11.6(i) Large , Fixed
… ►§11.6(ii) Large , Fixed
… ►More fully, the series (11.2.1) and (11.2.2) can be regarded as generalized asymptotic expansions (§2.1(v)). …9: 12.11 Zeros
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§12.11(ii) Asymptotic Expansions of Large Zeros
… ►When the zeros are asymptotically given by and , where is a large positive integer and … ►§12.11(iii) Asymptotic Expansions for Large Parameter
►For 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). … ►
12.11.9
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