relation to Airy function

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21: Bibliography S

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The Laplace transforms of products of Airy functions. Dirāsāt Ser. B Pure Appl. Sci. 19 (2), pp. 7–11.

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External Links:

ISSN 0253-424X, MathReview

Referenced by:

§9.10(v).

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The zeros and maxima of the Airy function and its first derivative to 25 significant figures. Report AFCRC-TR-59-135, ASTIA Document No. AD214568 Air Research and Development Command, U.S. Air Force, Bedford, MA.

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Notes:

Available from U.S. Department of Commerce, Office of Technical Services, Washington, D.C.

Referenced by:

2nd item, §9.8(iv).

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Structure of avoided crossings for eigenvalues related to equations of Heun’s class. J. Phys. A 30 (2), pp. 673–687.

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ISSN 0305-4470, Document, MathReview (Éric Delabaere), ZentralBlatt

Referenced by:

§31.13.

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Tables of Airy Functions and Special Confluent Hypergeometric Functions. Pergamon Press, New York.

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Notes:

Translated from the Russian by D. G. Fry

External Links:

MathReview (W. F. Freiberger), ZentralBlatt

Referenced by:

(9.13.19), (9.13.20), (9.13.21), §9.13(i), 1st item.

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A Riccati approach to the Airy equation. In Asymptotic and computational analysis (Winnipeg, MB, 1989), R. Wong (Ed.), pp. 403–415.

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External Links:

MathReview (Donald A. Lutz), ZentralBlatt

Referenced by:

§9.2(vi).

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22: 18.34 Bessel Polynomials

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§18.34(i) Definitions and Recurrence Relation

… ►where ${𝗄}_n$ is a modified spherical Bessel function (10.49.9), and … … ►The full system satisfies orthogonality with respect to a (not positive definite) moment functional; see Evans et al. (1993, (2.7)) for the simple expression of the moments ${\mu }_n$. … ►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). …

23: Bibliography

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Supplement to a paper “On the intensity of light in the neighbourhood of a caustic”. Trans. Camb. Phil. Soc. 8, pp. 595–599.

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Referenced by:

§9.16.

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Integrals involving Airy functions. J. Phys. A 19 (13), pp. 2663–2665.

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External Links:

ISSN 0305-4470, Document, MathReview, ZentralBlatt

Referenced by:

(9.11.11), (9.11.12), (9.11.13), (9.11.14), §9.11(iv).

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Integrals of products of Airy functions. J. Phys. A 10 (4), pp. 485–490.

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Document, MathReview (Y. L. Luke), ZentralBlatt

Referenced by:

(9.11.10), (9.11.5), (9.11.6), (9.11.7), (9.11.8), (9.11.9), §9.11(iv), §9.11(iv).

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Dirichlet series related to the Riemann zeta function. J. Number Theory 19 (1), pp. 85–102.

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ISSN 0022-314X, Document, MathReview (Kai Man Tsang), ZentralBlatt

Referenced by:

(25.16.10), (25.16.11), (25.16.12), (25.16.14), (25.16.15), (25.16.5), (25.16.6), (25.16.7), (25.16.8), (25.16.9), §25.16(ii), §25.16(ii), §25.16.

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Integral equations and relations for Lamé functions. Quart. J. Math. Oxford Ser. (2) 15, pp. 103–115.

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External Links:

Document, MathReview (I. Marx), ZentralBlatt

Referenced by:

§29.8.

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24: Bibliography Z

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On the Computation of Zeros of Bessel and Bessel-related Functions. In Proceedings of the Sixth International Colloquium on Differential Equations (Plovdiv, Bulgaria, 1995), D. Bainov (Ed.), Utrecht, pp. 409–416.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§10.74(vi).

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The Dilogarithm Function in Geometry and Number Theory. In Number Theory and Related Topics (Bombay, 1988), R. Askey and others (Eds.), Tata Inst. Fund. Res. Stud. Math., Vol. 12, pp. 231–249.

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External Links:

MathReview (J. Browkin), ZentralBlatt

Referenced by:

§25.12(i).

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Integrali di funzioni di Anger, Weber ed Airy-Hardy. Rend. Sem. Mat. Univ. Padova 58, pp. 275–285 (Italian).

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External Links:

FullText, MathReview (M. E. Muldoon), ZentralBlatt

Referenced by:

§11.10(x).

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Numerical analysis of Struve functions with applications to other special functions. Ann. Numer. Math. 2 (1-4), pp. 199–208.

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ISSN 1021-2655, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§11.7(ii).

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Generalized Watson Transforms and Applications to Group Representations. Ph.D. Thesis, University of Vermont, Burlington,VT.

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Referenced by:

§35.7(ii).

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25: 9.11 Products

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§9.11(i) Differential Equation

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§9.11(ii) Wronskian

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§9.11(iii) Integral Representations

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§9.11(iv) Indefinite Integrals

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§9.11(v) Definite Integrals

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26: 9.7 Asymptotic Expansions

§9.7 Asymptotic Expansions

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§9.7(iii) Error Bounds for Real Variables

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§9.7(iv) Error Bounds for Complex Variables

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§9.7(v) Exponentially-Improved Expansions

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27: 36.13 Kelvin’s Ship-Wave Pattern

§36.13 Kelvin’s Ship-Wave Pattern

… ►Then with $g$ denoting the acceleration due to gravity, the wave height is approximately given by … ►When $\rho >1$, that is, everywhere except close to the ship, the integrand oscillates rapidly. …The wake is a caustic of the “rays” defined by the dispersion relation (“Hamiltonian”) giving the frequency $\omega$ as a function of wavevector $𝐤$: … ►Then with the definitions (36.12.12), and the real functions …

28: Errata

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Specific source citations and proof metadata are now given for all equations in Chapter 25 Zeta and Related Functions.

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The Gegenbauer function $C_{\alpha }^{(\lambda )}\left(z\right)$, was labeled inadvertently as the ultraspherical (Gegenbauer) polynomial $C_n^{(\lambda )}\left(z\right)$. In order to resolve this inconsistency, this function now links correctly to its definition. This change affects Gegenbauer functions which appear in §§14.3(iv), 15.9(iii).

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The Kronecker delta symbols have been moved furthest to the right, as is common convention for orthogonality relations.

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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 number of additions and changes have been made to the metadata to reflect new and changed references as well as to how some equations have been derived.

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29: 28.8 Asymptotic Expansions for Large $q$

… ►For recurrence relations for the coefficients in these expansions see Frenkel and Portugal (2001, §3). … ►For recurrence relations for the coefficients in these expansions see Frenkel and Portugal (2001, §4 and §5). … ►Then as $h\to +\mathrm{\infty }$ … ►The approximants are elementary functions, Airy functions, Bessel functions, and parabolic cylinder functions; compare §2.8. … ►For related results see Langer (1934) and Sharples (1967, 1971). …

30: Bibliography T

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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).

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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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Erratum to “COULCC: A continued-fraction algorithm for Coulomb functions of complex order with complex arguments”. Comput. Phys. Comm. 159 (3), pp. 241–242.

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External Links:

Document, Code, ZentralBlatt

Referenced by:

§33.23(vi), I. J. Thompson and A. R. Barnett (1985).

Encodings:

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Hyperspherical elliptic harmonics and their relation to the Heun equation. Phys. Rev. A 63 (032510), pp. 1–8.

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External Links:

Document

Referenced by:

§31.17(ii).

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Level-spacing distributions and the Airy kernel. Comm. Math. Phys. 159 (1), pp. 151–174.

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External Links:

ISSN 0010-3616, FullText, MathReview (Estelle L. Basor), ZentralBlatt

Referenced by:

§32.14.

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