uniform asymptotic expansions for large order

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31: 2.1 Definitions and Elementary Properties

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§2.1(i) Asymptotic and Order Symbols

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§2.1(iii) Asymptotic Expansions

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§2.1(iv) Uniform Asymptotic Expansions

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§2.1(v) Generalized Asymptotic Expansions

… ►As in §2.1(iv), generalized asymptotic expansions can also have uniformity properties with respect to parameters. …

32: 13.21 Uniform Asymptotic Approximations for Large $\kappa$

§13.21 Uniform Asymptotic Approximations for Large $\kappa$

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§13.21(iv) Large $\kappa$, Other Expansions

►For a uniform asymptotic expansion in terms of Airy functions for $W_{\kappa ,\mu }\left(4\kappa x\right)$ when $\kappa$ is large and positive, $\mu$ is real with $|\mu |$ bounded, and $x\in [\delta ,\mathrm{\infty })$ see Olver (1997b, Chapter 11, Ex. 7.3). … ►For asymptotic expansions having double asymptotic properties see Skovgaard (1966). … ►

33: 10.74 Methods of Computation

… ►If $x$ or $|z|$ is large compared with ${|\nu |}^2$, then the asymptotic expansions of §§10.17(i)–10.17(iv) are available. … ►For large positive real values of $\nu$ the uniform asymptotic expansions of §§10.20(i) and 10.20(ii) can be used. Moreover, because of their double asymptotic properties (§10.41(v)) these expansions can also be used for large $x$ or $|z|$, whether or not $\nu$ is large. … ►In the case of the spherical Bessel functions the explicit formulas given in §§10.49(i) and 10.49(ii) are terminating cases of the asymptotic expansions given in §§10.17(i) and 10.40(i) for the Bessel functions and modified Bessel functions. … ►Methods for obtaining initial approximations to the zeros include asymptotic expansions (§§10.21(vi)-10.21(ix)), graphical intersection of $2D$ graphs in $ℝ$ (e. …

34: 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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ISSN 0036-1399, Document, MathReview (F. W. J. Olver), ZentralBlatt

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§10.45.

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Uniform asymptotic expansions of integrals with stationary point near algebraic singularity. Comm. Pure Appl. Math. 19, pp. 353–370.

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Document, MathReview (J. G. van der Corput), ZentralBlatt

Referenced by:

§2.3(v).

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Uniform asymptotic expansions of integrals with many nearby stationary points and algebraic singularities. J. Math. Mech. 17, pp. 533–559.

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MathReview (A. Erdélyi), ZentralBlatt

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§36.12(iii).

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Uniform asymptotic expansion of Charlier polynomials. Methods Appl. Anal. 1 (3), pp. 294–313.

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ISSN 1073-2772, Pdf, MathReview (Eberhard Wagner), ZentralBlatt

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§18.24.

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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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ISSN 0305-4470, Document, MathReview, ZentralBlatt

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§31.13.

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35: 2.10 Sums and Sequences

… ►for large $n$. … ►This identity can be used to find asymptotic approximations for large $n$ when the factor $v_j$ changes slowly with $j$, and $u_j$ is oscillatory; compare the approximation of Fourier integrals by integration by parts in §2.3(i). … ►

§2.10(iii) Asymptotic Expansions of Entire Functions

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Example

… ►For uniform expansions when two singularities coalesce on the circle of convergence see Wong and Zhao (2005). …

36: Bibliography S

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Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

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Document, MathReview (F. W. J. Olver), ZentralBlatt

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§28.8(iv).

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Uniform asymptotic expansions of modified Mathieu functions. J. Reine Angew. Math. 247, pp. 1–17.

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MathReview (J. Meixner), ZentralBlatt

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§28.8(iv).

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Error bounds for a uniform asymptotic expansion of the Legendre function $P_n^{-m}(\cosh z)$ . Quart. Appl. Math. 46 (3), pp. 473–488.

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ISSN 0033-569X, MathReview (F. W. J. Olver), ZentralBlatt

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§14.26.

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Uniform Asymptotic Expansions of Confluent Hypergeometric Functions and Whittaker Functions. Doctoral dissertation, University of Copenhagen, Vol. 1965, Jul. Gjellerups Forlag, Copenhagen.

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MathReview (A. K. Agarwal), ZentralBlatt

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§13.21(iv).

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Uniform asymptotic expansions of Hermite polynomials. M. Phil. thesis, City University of Hong Kong.

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Abstract

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§18.16(v).

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37: 2.4 Contour Integrals

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§2.4(i) Watson’s Lemma

… ►Then … ►For examples and extensions (including uniformity and loop integrals) see Olver (1997b, Chapter 4), Wong (1989, Chapter 1), and Temme (1985). … ►For large $t$, the asymptotic expansion of $q(t)$ may be obtained from (2.4.3) by Haar’s method. This depends on the availability of a comparison function $F(z)$ for $Q(z)$ that has an inverse transform …For examples see Olver (1997b, pp. 315–320). …

38: Bibliography C

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Asymptotic behaviour of the zeros of the (generalized) Laguerre polynomial $L_n^{\alpha }(x)$ as the index $\alpha \to \mathrm{\infty }$ and limiting formula relating Laguerre polynomials of large index and large argument to Hermite polynomials. Lett. Nuovo Cimento (2) 23 (3), pp. 101–102.

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ISSN 0024-1318, Document, MathReview (R. A. Askey)

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§18.7(iii).

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Asymptotic expansion of the first elliptic integral. SIAM J. Math. Anal. 16 (5), pp. 1072–1092.

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ISSN 0036-1410, Document, MathReview (Kusum Soni), ZentralBlatt

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§19.27(ii), §19.27(vi), §19.9(ii), §2.5(ii).

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Uniform asymptotic expansion of an integral with a saddle point, a pole and a branch point. Proc. Roy. Soc. London Ser. A 426, pp. 273–286.

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ISSN 0080-4630, FullText, MathReview, ZentralBlatt

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§2.4(vi).

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On the asymptotic expansion of Airy’s integral. Proc. Glasgow Math. Assoc. 6, pp. 113–115.

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Document, MathReview (L. Berg), ZentralBlatt

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(9.5.7), §9.5(ii).

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Asymptotic Expansions. Cambridge Tracts in Mathematics and Mathematical Physics, Cambridge University Press, New York.

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MathReview (A. E. Danese), ZentralBlatt

Referenced by:

Ch.2.

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