asymptotic%20approximations%20for%20large%20parameters
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1: 28.8 Asymptotic Expansions for Large
§28.8 Asymptotic Expansions for Large
… ►Barrett’s Expansions
… ►The approximations apply when the parameters and are real and large, and are uniform with respect to various regions in the -plane. … ►Dunster’s Approximations
… ►2: 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
… ►3: Bibliography D
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Uniform asymptotic expansions for prolate spheroidal functions with large parameters.
SIAM J. Math. Anal. 17 (6), pp. 1495–1524.
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Uniform asymptotic expansions for Whittaker’s confluent hypergeometric functions.
SIAM J. Math. Anal. 20 (3), pp. 744–760.
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Uniform asymptotic approximation of Mathieu functions.
Methods Appl. Anal. 1 (2), pp. 143–168.
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Uniform asymptotic expansions for associated Legendre functions of large order.
Proc. Roy. Soc. Edinburgh Sect. A 133 (4), pp. 807–827.
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Uniform asymptotic approximations for incomplete Riemann zeta functions.
J. Comput. Appl. Math. 190 (1-2), pp. 339–353.
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4: 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). … ►§30.9(iii) Other Approximations and Expansions
… ►5: Bibliography N
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The resurgence properties of the large order asymptotics of the Anger-Weber function I.
J. Class. Anal. 4 (1), pp. 1–39.
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The resurgence properties of the large order asymptotics of the Anger-Weber function II.
J. Class. Anal. 4 (2), pp. 121–147.
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On the large argument asymptotics of the Lommel function via Stieltjes transforms.
Asymptot. Anal. 91 (3-4), pp. 265–281.
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Error Bounds for the Large-Argument Asymptotic Expansions of the Hankel and Bessel Functions.
Acta Appl. Math. 150, pp. 141–177.
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Error bounds for the large-argument asymptotic expansions of the Lommel and allied functions.
Stud. Appl. Math. 140 (4), pp. 508–541.
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6: 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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7: 11.6 Asymptotic Expansions
§11.6 Asymptotic Expansions
►§11.6(i) Large , Fixed
… ►§11.6(ii) Large , Fixed
… ► … ►and for an estimate of the relative error in this approximation see Watson (1944, p. 336).8: Bibliography K
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Asymptotic approximations for the first incomplete elliptic integral near logarithmic singularity.
J. Comput. Appl. Math. 205 (1), pp. 186–206.
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Uniform asymptotic approximations for the Meixner-Sobolev polynomials.
Anal. Appl. (Singap.) 10 (3), pp. 345–361.
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Exponentially accurate uniform asymptotic approximations for integrals and Bleistein’s method revisited.
Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 469 (2153), pp. 20130008, 12.
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Connection formulae for asymptotics of solutions of the degenerate third Painlevé equation. I.
Inverse Problems 20 (4), pp. 1165–1206.
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Approximation Formulae for Generalized Hypergeometric Functions for Large Values of the Parameters.
J. B. Wolters, Groningen.
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9: 3.8 Nonlinear Equations
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►for all sufficiently large, where and are independent of , then the sequence is said to have convergence of the
th order.
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►Inverse linear interpolation (§3.3(v)) is used to obtain the first approximation:
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►Initial approximations to the zeros can often be found from asymptotic or other approximations to , or by application of the phase principle or Rouché’s theorem; see §1.10(iv).
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►Consider and .
We have and .
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10: 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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