modified expansions in terms of Airy functions

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1: 12.10 Uniform Asymptotic Expansions for Large Parameter

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Modified Expansions

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2: 10.72 Mathematical Applications

… ►In regions in which (10.72.1) has a simple turning point $z_0$, that is, $f(z)$ and $g(z)$ are analytic (or with weaker conditions if $z=x$ is a real variable) and $z_0$ is a simple zero of $f(z)$, asymptotic expansions of the solutions $w$ for large $u$ can be constructed in terms of Airy functions or equivalently Bessel functions or modified Bessel functions of order $\frac{1}{3}$ (§9.6(i)). …

3: 10.41 Asymptotic Expansions for Large Order

§10.41 Asymptotic Expansions for Large Order

… ► … ► … ►Similar analysis can be developed for the uniform asymptotic expansions in terms of Airy functions given in §10.20. … ►

4: 33.20 Expansions for Small $|ϵ|$

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§33.20(i) Case $ϵ=0$

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§33.20(ii) Power-Series in $ϵ$ for the Regular Solution

… ►The functions $Y$ and $K$ are as in §§10.2(ii), 10.25(ii), and the coefficients $C_{k,p}$ are given by (33.20.6). ►

§33.20(iv) Uniform Asymptotic Expansions

… ►These expansions are in terms of elementary functions, Airy functions, and Bessel functions of orders $2\mathrm{\ell }+1$ and $2\mathrm{\ell }+2$.

5: 2.8 Differential Equations with a Parameter

… ►These are elementary functions in Case I, and Airy functions (§9.2) in Case II. … ►For other examples of uniform asymptotic approximations and expansions of special functions in terms of Airy functions see especially §10.20 and §§12.10(vii), 12.10(viii); also §§12.14(ix), 13.20(v), 13.21(iii), 13.21(iv), 15.12(iii), 18.15(iv), 30.9(i), 30.9(ii), 32.11(ii), 32.11(iii), 33.12(i), 33.12(ii), 33.20(iv), 36.12(ii), 36.13. … ►For other examples of uniform asymptotic approximations and expansions of special functions in terms of Bessel functions or modified Bessel functions of fixed order see §§13.8(iii), 13.21(i), 13.21(iv), 14.15(i), 14.15(iii), 14.20(vii), 15.12(iii), 18.15(i), 18.15(iv), 18.24, 33.20(iv). … ►For further examples of uniform asymptotic approximations in terms of parabolic cylinder functions see §§13.20(iii), 13.20(iv), 14.15(v), 15.12(iii), 18.24. ►For further examples of uniform asymptotic approximations in terms of Bessel functions or modified Bessel functions of variable order see §§13.21(ii), 14.15(ii), 14.15(iv), 14.20(viii), 30.9(i), 30.9(ii). …

6: 28.8 Asymptotic Expansions for Large $q$

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§28.8(ii) Sips’ Expansions

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Barrett’s Expansions

… ►The approximants are elementary functions, Airy functions, Bessel functions, and parabolic cylinder functions; compare §2.8. … ►With additional restrictions on $z$, uniform asymptotic approximations for solutions of (28.2.1) and (28.20.1) are also obtained in terms of elementary functions by re-expansions of the Whittaker functions; compare §2.8(ii). ►Subsequently the asymptotic solutions involving either elementary or Whittaker functions are identified in terms of the Floquet solutions ${\mathrm{me}}_{\nu }(z,q)$ (§28.12(ii)) and modified Mathieu functions ${\mathrm{M}}_{\nu }^{(j)}(z,h)$ (§28.20(iii)). …

7: 18.32 OP’s with Respect to Freud Weights

… ►A Freud weight is a weight function of the form …These conditions on $Q(x)$ have been strengthened and also relaxed in literature. …However, for asymptotic approximations in terms of elementary functions for the OP’s, and also for their largest zeros, see Levin and Lubinsky (2001) and Nevai (1986). For a uniform asymptotic expansion in terms of Airy functions (§9.2) for the OP’s in the case $Q(x)=x^4$ see Bo and Wong (1999). … ►All of these forms appear in applications, see §18.39(iii) and Table 18.39.1, albeit sometimes with $x\in [0,\mathrm{\infty })$, where the term half-Freud weight is used; or on $x\in [-1,1]$ or $[0,1]$, where the term Rys weight is employed, see Rys et al. (1983). …

8: Bibliography T

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On the numerical evaluation of the modified Bessel function of the third kind. J. Comput. Phys. 19 (3), pp. 324–337.

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Notes:

Algol program.

External Links:

Document, MathReview (Carl C. Grosjean), ZentralBlatt

Referenced by:

§10.74(i), §10.74(iv), §10.77(iii).

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The asymptotic expansion of the incomplete gamma functions. SIAM J. Math. Anal. 10 (4), pp. 757–766.

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External Links:

ISSN 0036-1410, Document, MathReview (E. Riekstiņš), ZentralBlatt

Referenced by:

§8.12.

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Uniform asymptotic expansions of a class of integrals in terms of modified Bessel functions, with application to confluent hypergeometric functions. SIAM J. Math. Anal. 21 (1), pp. 241–261.

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External Links:

ISSN 0036-1410, Document, MathReview (Pablo Martín), ZentralBlatt

Referenced by:

§13.8(iii), §13.8(iii).

Encodings:

BibTeX

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Numerical algorithms for uniform Airy-type asymptotic expansions. Numer. Algorithms 15 (2), pp. 207–225.

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External Links:

ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§10.19(iii), §10.20(i), §10.74(i).

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The numerical computation of special functions by use of quadrature rules for saddle point integrals. II. Gamma functions, modified Bessel functions and parabolic cylinder functions. Report TW 183/78 Mathematisch Centrum, Amsterdam, Afdeling Toegepaste Wiskunde.

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Notes:

Algol procedures, maximum accuracy 14S.

External Links:

FullText, ZentralBlatt

Referenced by:

§12.21(ii).

Encodings:

BibTeX

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9: Bibliography S

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On the expansion of the parabolic cylinder function in a series of the product of two parabolic cylinder functions. J. Indian Math. Soc. (N. S.) 3, pp. 226–230.

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External Links:

ZentralBlatt

Referenced by:

§12.13(i).

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BibTeX

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

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External Links:

MathReview (J. Meixner), ZentralBlatt

Referenced by:

§28.8(iv).

Encodings:

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Characterization of Jacobian varieties in terms of soliton equations. Invent. Math. 83 (2), pp. 333–382.

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ISSN 0020-9910, Document, MathReview (Bert van Geemen), ZentralBlatt

Referenced by:

§21.9.

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On the calculation of complex zeros of the modified Bessel function of the second kind. Dokl. Akad. Nauk SSSR 280 (2), pp. 296–299.

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Notes:

English translation in Soviet Math. Dokl. 31(1), pp. 78–81

External Links:

ISSN 0002-3264, MathReview (I. F. Kushnirchuk), ZentralBlatt

Referenced by:

§10.74(vi).

Encodings:

BibTeX

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Exact error terms in the asymptotic expansion of a class of integral transforms. I. Oscillatory kernels. SIAM J. Math. Anal. 11 (5), pp. 828–841.

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External Links:

ISSN 0036-1410, Document, MathReview (John S. Lew), ZentralBlatt

Referenced by:

§2.5(ii).

Encodings:

BibTeX

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10: Bibliography B

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Zeros of generalized Airy functions. Mathematika 32 (1), pp. 104–117.

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External Links:

ISSN 0025-5793, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§9.13(ii).

Encodings:

BibTeX

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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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External Links:

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

Referenced by:

§10.45.

Encodings:

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Tables of modified Struve functions of orders zero and unity.

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Notes:

Part of the Unpublished Mathematical Tables. Reviewed in Math. Comp., v. 18 (1964), no. 86, p. 332.

Referenced by:

3rd item.

Encodings:

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Asymptotic expansions for the coefficient functions that arise in turning-point problems. Proc. Roy. Soc. London Ser. A 410, pp. 35–60.

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External Links:

ISSN 0080-4630, FullText, MathReview (P. F. Hsieh), ZentralBlatt

Referenced by:

§2.8(iii).

Encodings:

BibTeX

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Asymptotic Expansions for the Coefficient Functions Associated with Linear Second-order Differential Equations: The Simple Pole Case. In Asymptotic and Computational Analysis (Winnipeg, MB, 1989), R. Wong (Ed.), Lecture Notes in Pure and Applied Mathematics, Vol. 124, pp. 53–73.

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External Links:

MathReview (John S. Lew), ZentralBlatt

Referenced by:

§2.8(iv).

Encodings:

BibTeX

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