modified expansions in terms of Airy functions

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11: Bibliography L

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Microwave specific attenuation by oblate spheroidal raindrops: An exact analysis of TCS’s in terms of spheroidal wave functions. J. Electromagn. Waves Appl. 12 (6), pp. 709–711.

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ISSN 0920-5071, Document, MathReview

Referenced by:

§30.14(v).

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Uniform approximations of Bernoulli and Euler polynomials in terms of hyperbolic functions. Stud. Appl. Math. 103 (3), pp. 241–258.

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ISSN 0022-2526, Document, Link, MathReview, ZentralBlatt

Referenced by:

§24.11, §24.11.

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Monotonicity in terms of order of the zeros of the derivatives of Bessel functions. Proc. Amer. Math. Soc. 108 (2), pp. 387–389.

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ISSN 0002-9939, FullText, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§10.21(iv), §10.21(iv).

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Integral representation of the Hankel function in terms of parabolic cylinder functions. Quart. J. Mech. Appl. Math. 23 (3), pp. 315–327.

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Document, MathReview (T. Erber), ZentralBlatt

Referenced by:

§12.12, §12.16.

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Expansion of the confluent hypergeometric function in series of Bessel functions. Math. Tables Aids Comput. 13 (68), pp. 261–271.

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FullText, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§13.24(ii).

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12: Errata

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18.34.2

$y_n(x)=y_n(x;2)=2{\pi }^{-1}{x}^{-1}{\mathrm{e}}^{1/x}{𝗄}_n\left(x^{-1}\right),$

${\theta }_n(x)=x^n{y}_n(x^{-1})=2{\pi }^{-1}{x}^{n+1}{\mathrm{e}}^x{𝗄}_n\left(x\right)$

This equation was updated to include definitions in terms of the modified spherical Bessel function of the second kind.

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§4.13 has been enlarged. The Lambert $W$-function is multi-valued and we use the notation $W_k\left(x\right)$, $k\in \mathbb{Z}$, for the branches. The original two solutions are identified via $\mathrm{Wp}\left(x\right)=W_0\left(x\right)$ and $\mathrm{Wm}\left(x\right)=W_{\pm 1}\left(x\mp 0\mathrm{i}\right)$.

Other changes are the introduction of the Wright $\omega$-function and tree $T$-function in (4.13.1_2) and (4.13.1_3), simplification formulas (4.13.3_1) and (4.13.3_2), explicit representation (4.13.4_1) for $\frac{d^{n}W}{{dz}^n}$, additional Maclaurin series (4.13.5_1) and (4.13.5_2), an explicit expansion about the branch point at $z=-{\mathrm{e}}^{-1}$ in (4.13.9_1), extending the number of terms in asymptotic expansions (4.13.10) and (4.13.11), and including several integrals and integral representations for Lambert $W$-functions in the end of the section.

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The validity constraint was added. Additionally, specific source citations are now given in the metadata for all equations in Chapter 9 Airy and Related Functions.

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A short paragraph dealing with asymptotic approximations that are expressed in terms of two or more Poincaré asymptotic expansions has been added below (2.1.16).

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Originally was expressed in term of asymptotic symbol $\sim$. As a consequence of the use of the $O$ order symbol on the right-hand side, $\sim$ was replaced by $=$.

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13: 9.12 Scorer Functions

§9.12 Scorer Functions

… ► $-\mathrm{Gi}\left(x\right)$ is a numerically satisfactory companion to the complementary functions $\mathrm{Ai}\left(x\right)$ and $\mathrm{Bi}\left(x\right)$ on the interval . … ►

§9.12(viii) Asymptotic Expansions

… ►For other phase ranges combine these results with the connection formulas (9.12.11)–(9.12.14) and the asymptotic expansions given in §9.7. … ►

Integrals

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14: 18.15 Asymptotic Approximations

… ►These expansions are in terms of Whittaker functions (§13.14). … ►These expansions are in terms of Bessel functions and modified Bessel functions, respectively. … ►

In Terms of Elementary Functions

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In Terms of Airy Functions

… ►For an error bound for the first term in the Airy-function expansions see Olver (1997b, p. 403). …

15: Bibliography C

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Expansions in terms of parabolic cylinder functions. Proc. Edinburgh Math. Soc. (2) 8, pp. 50–65.

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External Links:

MathReview (I. I. Hirschman, Jr.), ZentralBlatt

Referenced by:

§12.16.

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A dispersion analysis for difference schemes: Tables of generalized Airy functions. Math. Comp. 32 (144), pp. 1163–1170.

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ISSN 0025-5718, FullText, MathReview (Philip Brenner), ZentralBlatt

Referenced by:

§9.13(ii).

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Chebyshev expansions for the Bessel function $J_n(z)$ in the complex plane. Math. Comp. 40 (161), pp. 343–366.

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ISSN 0025-5718, FullText, MathReview (C. W. Clenshaw), ZentralBlatt

Referenced by:

§10.76(ii).

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Efficient and accurate computation of the incomplete Airy functions. Radio Science 28 (4), pp. 441–457.

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ISSN 0048-6604, Document

Referenced by:

§9.14.

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

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External Links:

Document, MathReview (L. Berg), ZentralBlatt

Referenced by:

(9.5.7), §9.5(ii).

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16: 2.4 Contour Integrals

… ►Now assume that $c>0$ and we are given a function $Q(z)$ that is both analytic and has the expansion … ►Thus the right-hand side of (2.4.14) reduces to the error terms. …In consequence, the asymptotic expansion obtained from (2.4.14) is no longer null. … ►and assigning an appropriate value to $c$ to modify the contour, the approximating integral is reducible to an Airy function or a Scorer function (§§9.2, 9.12). … ►For a symbolic method for evaluating the coefficients in the asymptotic expansions see Vidūnas and Temme (2002). …

17: Bibliography R

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High precision Chebyshev expansions for Airy functions and their derivatives. Technical report University of Birmingham Computer Centre.

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3rd item.

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Remark on Algorithm 498: Airy functions using Chebyshev series approximations. ACM Trans. Math. Software 7 (3), pp. 404–405.

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ISSN 0098-3500, Document

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1st item.

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Integral representations for products of Airy functions. Z. Angew. Math. Phys. 46 (2), pp. 159–170.

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External Links:

ISSN 0044-2275, Document, MathReview (P. K. Banerji), ZentralBlatt

Referenced by:

§9.11(iii), §9.11(v), §9.11(v), (9.5.6), §9.5(ii).

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Integral representations for products of Airy functions. II. Cubic products. Z. Angew. Math. Phys. 48 (4), pp. 646–655.

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External Links:

ISSN 0044-2275, Document, MathReview (E. R. Love), ZentralBlatt

Referenced by:

(9.11.16), (9.11.17), §9.11(iii), §9.11(v).

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Infinite Sum of the Incomplete Gamma Function Expressed in Terms of the Hurwitz Zeta Function. Mathematics 9 (16).

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Link, ISSN 2227-7390, Document

Referenced by:

§8.15.

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18: 10.21 Zeros

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§10.21(vi) McMahon’s Asymptotic Expansions for Large Zeros

… ►For the next three terms in (10.21.19) and the next two terms in (10.21.20) see Bickley et al. (1952, p. xxxvii) or Olver (1960, pp. xvii–xviii). … ►Let ${𝒞}_{\nu }\left(x\right)$, ${\rho }_{\nu }(t)$, and ${\sigma }_{\nu }(t)$ be defined as in §10.21(ii) and $M\left(x\right)$, $\theta \left(x\right)$, $N\left(x\right)$, and $\varphi \left(x\right)$ denote the modulus and phase functions for the Airy functions and their derivatives as in §9.8. … ►Here $a_m$ and $a_m^{\prime }$ denote respectively the zeros of the Airy function $\mathrm{Ai}\left(z\right)$ and its derivative ${\mathrm{Ai}}^{\prime }\left(z\right)$; see §9.9. … ►Higher coefficients in the asymptotic expansions in this subsection can be obtained by expressing the cross-products in terms of the modulus and phase functions (§10.18), and then reverting the asymptotic expansion for the difference of the phase functions. …

19: 18.34 Bessel Polynomials

… ►where ${𝗄}_n$ is a modified spherical Bessel function (10.49.9), and … ►expressed in terms of Romanovski–Bessel polynomials, Laguerre polynomials or Whittaker functions, we have …In this limit the finite system of Jacobi polynomials $P_n^{(\alpha ,\beta )}\left(x\right)$ which is orthogonal on $(1,\mathrm{\infty })$ (see §18.3) tends to the finite system of Romanovski–Bessel polynomials which is orthogonal on $(0,\mathrm{\infty })$ (see (18.34.5_5)). ►For uniform asymptotic expansions of $y_n(x;a)$ as $n\to \mathrm{\infty }$ in terms of Airy functions (§9.2) see Wong and Zhang (1997) and Dunster (2001c). For uniform asymptotic expansions in terms of Hermite polynomials see López and Temme (1999b). …

20: Bibliography M

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Chebyshev expansions for modified Struve and related functions. Math. Comp. 60 (202), pp. 735–747.

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ISSN 0025-5718, FullText, MathReview, ZentralBlatt

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2nd item.

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Two-point quasi-fractional approximations to the Airy function $\mathrm{Ai}(x)$ . J. Comput. Phys. 99 (2), pp. 337–340.

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Notes:

There is an error in Eq. (5): the second term in parentheses needs to be multiplied by $x$.

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ISSN 0021-9991, Document, MathReview Entry, ZentralBlatt

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1st item.

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Explicit error terms for asymptotic expansions of Stieltjes transforms. J. Inst. Math. Appl. 22 (2), pp. 129–145.

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ISSN 0020-2932, Document, MathReview (G. M. Habibullah), ZentralBlatt

Referenced by:

§2.6(ii).

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The Second Painlevé Transcendent: A Nonlinear Airy Function. In Mechanics Today, Vol. 5, pp. 297–313.

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External Links:

MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§32.11(ii), §32.17, §32.3(ii).

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Asymptotic expansions of ellipsoidal wave functions in terms of Hermite functions. Math. Nachr. 32, pp. 49–62.

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External Links:

Document, MathReview (M. J. O. Strutt), ZentralBlatt

Referenced by:

§29.11, §29.7(ii).

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