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§12.10 Uniform Asymptotic Expansions for Large Parameter

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§12.10(vi) Modifications of Expansions in Elementary Functions

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Modified Expansions

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C. B. Balogh (1967) 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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External Links:: ISSN 0036-1399, Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §10.45.
Encodings:: BibTeX

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B. C. Berndt and R. J. Evans (1984) Chapter 13 of Ramanujan’s second notebook: Integrals and asymptotic expansions. Expo. Math. 2 (4), pp. 289–347.

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External Links:: ISSN 0723-0869, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §2.10(iii).
Encodings:: BibTeX

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N. Bleistein and R. A. Handelsman (1975) Asymptotic Expansions of Integrals. Holt, Rinehart, and Winston, New York.

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Notes:: Reprinted with corrections by Dover Publications Inc., New York, 1986
External Links:: ISBN 0-486-65082-0, MathReview, ZentralBlatt
Referenced by:: §2.3(ii), §2.3(v), §2.4(iv), §2.4(v), §2.5(iii), §2.5(ii), Ch.2, Ch.9.
Encodings:: BibTeX

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R. Bo and R. Wong (1994) Uniform asymptotic expansion of Charlier polynomials. Methods Appl. Anal. 1 (3), pp. 294–313.

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External Links:: ISSN 1073-2772, Pdf, MathReview (Eberhard Wagner), ZentralBlatt
Referenced by:: §18.24.
Encodings:: BibTeX

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J. Brüning (1984) On the asymptotic expansion of some integrals. Arch. Math. (Basel) 42 (3), pp. 253–259.

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External Links:: ISSN 0003-889X, Document, MathReview (E. Riekstiņš), ZentralBlatt
Referenced by:: §2.5(ii).
Encodings:: BibTeX

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B. R. Fabijonas and F. W. J. Olver (1999) On the reversion of an asymptotic expansion and the zeros of the Airy functions. SIAM Rev. 41 (4), pp. 762–773.

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External Links:: ISSN 1095-7200, Document, MathReview (Eberhard Wagner), ZentralBlatt
Referenced by:: §2.2, §2.2, §3.8(v), §9.9(iv), §9.9(iv).
Encodings:: BibTeX

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S. Farid Khwaja and A. B. Olde Daalhuis (2014) Uniform asymptotic expansions for hypergeometric functions with large parameters IV. Anal. Appl. (Singap.) 12 (6), pp. 667–710.

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External Links:: ISSN 0219-5305, Document, Link, MathReview (Javier Segura), ZentralBlatt
Referenced by:: §15.12(iii), §15.12(iii).
Encodings:: BibTeX

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C. Ferreira and J. L. López (2001) An asymptotic expansion of the double gamma function. J. Approx. Theory 111 (2), pp. 298–314.

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External Links:: ISSN 0021-9045, Document, MathReview (C. M. Joshi), ZentralBlatt
Referenced by:: §5.17.
Encodings:: BibTeX

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J. L. Fields and Y. L. Luke (1963a) 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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External Links:: Document, MathReview (L. J. Slater), ZentralBlatt
Referenced by:: §16.11(iii).
Encodings:: BibTeX

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C. L. Frenzen (1987b) On the asymptotic expansion of Mellin transforms. SIAM J. Math. Anal. 18 (1), pp. 273–282.

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External Links:: ISSN 0036-1410, Document, Link, MathReview (Archibald Brown), ZentralBlatt
Referenced by:: §2.3(vi).
Encodings:: BibTeX

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§30.9 Asymptotic Approximations and Expansions

… ►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). … ►

§30.9(iii) Other Approximations and Expansions

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28.8.1

\rselection{a_{m}\left(h^{2}\right)\\ b_{m+1}\left(h^{2}\right)}\sim-2h^{2}+2sh-\frac{1}{8}(s^{2}+1)-\frac{1}{2^{7}h% }(s^{3}+3s)-\frac{1}{2^{12}h^{2}}(5s^{4}+34s^{2}+9)-\frac{1}{2^{17}h^{3}}(33s^% {5}+410s^{3}+405s)-\frac{1}{2^{20}h^{4}}(63s^{6}+1260s^{4}+2943s^{2}+486)-% \frac{1}{2^{25}h^{5}}(527s^{7}+15617s^{5}+69001s^{3}+41607s)+\cdots.

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Symbols:: $a_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $b_{\NVar{n}}\left(\NVar{q}\right)$ : eigenvalues of Mathieu equation, $\sim$ : Poincaré asymptotic expansion, $m$ : integer and $h$ : parameter
A&S Ref:: 20.2.30 (in different form)
Referenced by:: §28.8(i), Erratum (V1.1.10) for Sections 28.6(i), 28.6(ii), 28.8(i), 28.8(ii)
Permalink:: http://dlmf.nist.gov/28.8.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §28.8(i), §28.8 and Ch.28

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Z. Wang and R. Wong (2003) Asymptotic expansions for second-order linear difference equations with a turning point. Numer. Math. 94 (1), pp. 147–194.

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External Links:: ISSN 0029-599X, Document, MathReview (Sergey M. Zagorodnyuk), ZentralBlatt
Referenced by:: §2.9(iii).
Encodings:: BibTeX

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R. Wong (1973b) On uniform asymptotic expansion of definite integrals. J. Approximation Theory 7 (1), pp. 76–86.

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External Links:: Document, MathReview (L. Berg), ZentralBlatt
Referenced by:: §8.11(v).
Encodings:: BibTeX

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R. Wong (1981) Asymptotic expansions of the Kontorovich-Lebedev transform. Appl. Anal. 12 (3), pp. 161–172.

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External Links:: ISSN 0003-6811, Document, MathReview (E. R. Love), ZentralBlatt
Referenced by:: §10.43(v).
Encodings:: BibTeX

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R. Wong (1983) Applications of some recent results in asymptotic expansions. Congr. Numer. 37, pp. 145–182.

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External Links:: ISSN 0384-9864, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §19.27(vi).
Encodings:: BibTeX

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E. M. Wright (1935) The asymptotic expansion of the generalized Bessel function. Proc. London Math. Soc. (2) 38, pp. 257–270.

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External Links:: Document, ZentralBlatt
Referenced by:: §10.46.
Encodings:: BibTeX

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§11.6 Asymptotic Expansions

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§11.6(i) Large $|z|$ , Fixed $\nu$

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§11.6(ii) Large $|\nu|$ , Fixed $z$

… ►More fully, the series (11.2.1) and (11.2.2) can be regarded as generalized asymptotic expansions (§2.1(v)). …

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A. J. MacLeod (2002a) Asymptotic expansions for the zeros of certain special functions. J. Comput. Appl. Math. 145 (2), pp. 261–267.

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External Links:: ISSN 0377-0427, Document, MathReview (Andrea Laforgia), ZentralBlatt
Referenced by:: §10.70, §11.4(vii), §6.13.
Encodings:: BibTeX

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J. P. McClure and R. Wong (1978) Explicit error terms for asymptotic expansions of Stieltjes transforms. J. Inst. Math. Appl. 22 (2), pp. 129–145.

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External Links:: ISSN 0020-2932, Document, MathReview (G. M. Habibullah), ZentralBlatt
Referenced by:: §2.6(ii).
Encodings:: BibTeX

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J. P. McClure and R. Wong (1979) Exact remainders for asymptotic expansions of fractional integrals. J. Inst. Math. Appl. 24 (2), pp. 139–147.

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External Links:: ISSN 0020-2932, Document, MathReview, ZentralBlatt
Referenced by:: §2.6(iii).
Encodings:: BibTeX

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J. P. McClure and R. Wong (1987) Asymptotic expansion of a multiple integral. SIAM J. Math. Anal. 18 (6), pp. 1630–1637.

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External Links:: ISSN 0036-1410, Document, MathReview (E. Riekstiņš), ZentralBlatt
Referenced by:: §2.5(ii).
Encodings:: BibTeX

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H. J. W. Müller (1966c) On asymptotic expansions of ellipsoidal wave functions. Math. Nachr. 32, pp. 157–172.

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External Links:: Document, MathReview (F. M. Arscott), ZentralBlatt
Referenced by:: §29.11, §29.7(ii).
Encodings:: BibTeX

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R. Vidūnas and N. M. Temme (2002) Symbolic evaluation of coefficients in Airy-type asymptotic expansions. J. Math. Anal. Appl. 269 (1), pp. 317–331.

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External Links:: ISSN 0022-247X, Document, MathReview, ZentralBlatt
Referenced by:: §2.4(v).
Encodings:: BibTeX

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H. Volkmer (1999) Expansions in products of Heine-Stieltjes polynomials. Constr. Approx. 15 (4), pp. 467–480.

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External Links:: ISSN 0176-4276, Document, MathReview (Luoqing Li), ZentralBlatt
Referenced by:: §31.15(iii).
Encodings:: BibTeX

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H. Volkmer and J. J. Wood (2014) A note on the asymptotic expansion of generalized hypergeometric functions. Anal. Appl. (Singap.) 12 (1), pp. 107–115.

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External Links:: ISSN 0219-5305, Document, MathReview (Roberto S. Costas-Santos), ZentralBlatt
Referenced by:: §16.11(ii), §16.11(ii).
Encodings:: BibTeX

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H. Volkmer (2004a) Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.

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External Links:: ISSN 0176-4276, Document, MathReview, ZentralBlatt
Referenced by:: §30.16(i), §30.16(ii), §30.16(ii).
Encodings:: BibTeX

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H. Volkmer (2023) Asymptotic expansion of the generalized hypergeometric function

{}_{p}F_{q}(z)

as

z\to\infty

for

p<q

. Anal. Appl. (Singap.) 21 (2), pp. 535–545.

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External Links:: ISSN 0219-5305, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §16.11(i), §16.11, Erratum (V1.1.9) for Section 16.11(i).
Encodings:: BibTeX

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A. B. Olde Daalhuis (1994) Asymptotic expansions for

q

-gamma,

q

-exponential, and

q

-Bessel functions. J. Math. Anal. Appl. 186 (3), pp. 896–913.

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External Links:: ISSN 0022-247X, Document, MathReview (P. K. Banerji), ZentralBlatt
Referenced by:: §5.18(ii).
Encodings:: BibTeX

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A. B. Olde Daalhuis (1998c) On the resurgence properties of the uniform asymptotic expansion of the incomplete gamma function. Methods Appl. Anal. 5 (4), pp. 425–438.

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External Links:: ISSN 1073-2772, Pdf, MathReview (Éric Delabaere), ZentralBlatt
Referenced by:: §8.12.
Encodings:: BibTeX

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A. B. Olde Daalhuis (2000) On the asymptotics for late coefficients in uniform asymptotic expansions of integrals with coalescing saddles. Methods Appl. Anal. 7 (4), pp. 727–745.

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External Links:: ISSN 1073-2772, Pdf, MathReview (Raimundas Vidunas), ZentralBlatt
Referenced by:: §36.12(iii).
Encodings:: BibTeX

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A. B. Olde Daalhuis (2003a) Uniform asymptotic expansions for hypergeometric functions with large parameters. I. Analysis and Applications (Singapore) 1 (1), pp. 111–120.

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External Links:: ISSN 0219-5305, Document, MathReview, ZentralBlatt
Referenced by:: §15.12(iii).
Encodings:: BibTeX

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F. W. J. Olver (1994a) Asymptotic expansions of the coefficients in asymptotic series solutions of linear differential equations. Methods Appl. Anal. 1 (1), pp. 1–13.

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External Links:: ISSN 1073-2772, Pdf, MathReview (Archibald D. MacGillivray), ZentralBlatt
Referenced by:: §2.7(ii), §2.7(ii).
Encodings:: BibTeX

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asymptotic%20expansions

11—20 of 27 matching pages

11: 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

12: Bibliography B

13: Bibliography F

14: 30.9 Asymptotic Approximations and Expansions

§30.9 Asymptotic Approximations and Expansions

§30.9(iii) Other Approximations and Expansions

15: 28.8 Asymptotic Expansions for Large $q$

16: Bibliography W

17: 11.6 Asymptotic Expansions

§11.6 Asymptotic Expansions

§11.6(i) Large $|z|$ , Fixed $\nu$

§11.6(ii) Large $|\nu|$ , Fixed $z$

18: Bibliography M

19: Bibliography V

20: Bibliography O

asymptotic%20expansions

11—20 of 27 matching pages

11: 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

12: Bibliography B

13: Bibliography F

14: 30.9 Asymptotic Approximations and Expansions

§30.9 Asymptotic Approximations and Expansions

§30.9(iii) Other Approximations and Expansions

15: 28.8 Asymptotic Expansions for Large q

16: Bibliography W

17: 11.6 Asymptotic Expansions

§11.6 Asymptotic Expansions

§11.6(i) Large | z | , Fixed ν

§11.6(ii) Large | ν | , Fixed z

18: Bibliography M

19: Bibliography V

20: Bibliography O

15: 28.8 Asymptotic Expansions for Large $q$

§11.6(i) Large $|z|$ , Fixed $\nu$

§11.6(ii) Large $|\nu|$ , Fixed $z$