relation to Airy function

SearchAdvancedHelp

(0.013 seconds)

11—20 of 46 matching pages

11: 10.39 Relations to Other Functions

§10.39 Relations to Other Functions

… ►

Airy Functions

… ►

Parabolic Cylinder Functions

… ►

Confluent Hypergeometric Functions

… ►

Generalized Hypergeometric Functions and Hypergeometric Function

…

12: Frank W. J. Olver

… ►Olver joined NIST in 1961 after having been recruited by Milton Abramowitz to be the author of the Chapter “Bessel Functions of Integer Order” in the Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, a publication which went on to become the most widely distributed and most highly cited publication in NIST’s history. … ►, the behavior of solutions as the independent variable, or some parameter, tends to infinity, and in the study of the particular solutions of differential equations known as special functions (e. …Having witnessed the birth of the computer age firsthand (as a colleague of Alan Turing at NPL, for example), Olver is also well known for his contributions to the development and analysis of numerical methods for computing special functions. … ►In April 2011, NIST co-organized a conference on “Special Functions in the 21st Century: Theory & Application” which was dedicated to Olver. … ►

9 Airy and Related Functions

…

13: Bibliography V

… ►

An integral transform involving Heun functions and a related eigenvalue problem. SIAM J. Math. Anal. 17 (3), pp. 688–703.

ⓘ

External Links:

ISSN 0036-1410, Document, MathReview (William L. Perry), ZentralBlatt

Referenced by:

§31.10(i).

Encodings:

BibTeX

►

Airy Functions and Applications to Physics. Second edition, Imperial College Press, London.

ⓘ

External Links:

ISBN 978-1-84816-548-9; 1-84816-548-X, MathReview Entry, ZentralBlatt

Referenced by:

§1.17(ii), §9.10(ii), §9.10(ix), §9.11(v), §9.16, §9.16, §9.5(i).

Encodings:

BibTeX

… ►

Symbolic evaluation of coefficients in Airy-type asymptotic expansions. J. Math. Anal. Appl. 269 (1), pp. 317–331.

ⓘ

External Links:

ISSN 0022-247X, Document, MathReview, ZentralBlatt

Referenced by:

§2.4(v).

Encodings:

BibTeX

… ►

Integral relations for Lamé functions. SIAM J. Math. Anal. 13 (6), pp. 978–987.

ⓘ

External Links:

ISSN 0036-1410, Document, MathReview (F. M. Arscott), ZentralBlatt

Referenced by:

§29.8, §29.8.

Encodings:

BibTeX

… ►

Asymptotic expansion of the generalized hypergeometric function ${}_p{}^{}F_q^{}(z)$ as $z\to \mathrm{\infty }$ for . Anal. Appl. (Singap.) 21 (2), pp. 535–545.

ⓘ

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

…

14: 9 Airy and Related Functions

Chapter 9 Airy and Related Functions

…

15: Bibliography N

… ►

Integrals of Airy Functions. National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..

ⓘ

External Links:

MathReview (J. C. P. Miller), ZentralBlatt

Referenced by:

2nd item, 3rd item.

Encodings:

BibTeX

►

Tables Relating to Mathieu Functions: Characteristic Values, Coefficients, and Joining Factors. 2nd edition, National Bureau of Standards Applied Mathematics Series, U.S. Government Printing Office, Washington, D.C..

ⓘ

External Links:

MathReview, ZentralBlatt

Referenced by:

§28.1, §28.26(i), 6th item, Notations S, Notations S.

Encodings:

BibTeX

►

Confluent hypergeometric equations and related solvable potentials in quantum mechanics. J. Math. Phys. 41 (12), pp. 7964–7996.

ⓘ

External Links:

ISSN 0022-2488, Document, MathReview, ZentralBlatt

Referenced by:

§13.28(i).

Encodings:

BibTeX

… ►

Proofs of two conjectures on the real zeros of the cylinder and Airy functions. SIAM J. Math. Anal. 53 (4), pp. 4328–4349.

ⓘ

External Links:

ISSN 0036-1410, Document, Link, MathReview (José Luis López), ZentralBlatt

Referenced by:

(10.18.17), (10.18.18), §10.18(iii), (9.8.22), (9.8.23), §9.8(iv), Erratum (V1.1.8) for Section 9.8(iv), Erratum (V1.1.8) for Section 10.18(iii).

Encodings:

BibTeX

… ►

Tables of Generalized Airy Functions for the Asymptotic Solution of the Differential Equations $ϵ{(p{y}^{\prime })}^{\prime }+(q+ϵr)y=f$ . Pergamon Press, Oxford.

ⓘ

Notes:

D. E. Brown, translator.

External Links:

MathReview, ZentralBlatt

Referenced by:

4th item.

Encodings:

BibTeX

…

16: 16.18 Special Cases

§16.18 Special Cases

… ►This is a consequence of the following relations: …As a corollary, special cases of the ${}_1{}^{}F_1^{}$ and ${}_2{}^{}F_1^{}$ functions, including Airy functions, Bessel functions, parabolic cylinder functions, Ferrers functions, associated Legendre functions, and many orthogonal polynomials, are all special cases of the Meijer $G$-function. Representations of special functions in terms of the Meijer $G$-function are given in Erdélyi et al. (1953a, §5.6), Luke (1969a, §§6.4–6.5), and Mathai (1993, §3.10).

17: Bibliography G

… ►

Some elementary inequalities relating to the gamma and incomplete gamma function. J. Math. Phys. 38 (1), pp. 77–81.

ⓘ

External Links:

MathReview (H. O. Pollak), ZentralBlatt

Referenced by:

§5.6(i), §6.8, §7.8, §8.10.

Encodings:

BibTeX

… ►

Questions of Numerical Condition Related to Polynomials. In Studies in Numerical Analysis, G. H. Golub (Ed.), pp. 140–177.

ⓘ

External Links:

MathReview Entry, ZentralBlatt

Referenced by:

§3.8(vi).

Encodings:

BibTeX

… ►

Computation of Bessel and Airy functions and of related Gaussian quadrature formulae. BIT 42 (1), pp. 110–118.

ⓘ

External Links:

ISSN 0006-3835, Document, MathReview (Khélifa Trimèche), ZentralBlatt

Referenced by:

§10.74(iii), §9.17(iii).

Encodings:

BibTeX

… ►

Problem 72-21, Laplace transforms of Airy functions. SIAM Rev. 15 (4), pp. 796–798.

ⓘ

Notes:

Problem proposed by P. Smith.

External Links:

Document

Referenced by:

(9.10.14), (9.10.15), (9.10.16), §9.10(v).

Encodings:

BibTeX

… ►

An application of Airy functions to the Tricomi problem. Math. Nachr. 102 (1), pp. 169–181.

ⓘ

External Links:

ISSN 0025-584X, Document, MathReview (John M. S. Rassias), ZentralBlatt

Referenced by:

§9.16.

Encodings:

BibTeX

…

18: 1.17 Integral and Series Representations of the Dirac Delta

… ►subject to certain conditions on the function $\varphi (x)$. … ►

Coulomb Functions (§33.14(iv))

… ►

Airy Functions (§9.2)

… ►In the language of physics and applied mathematics, these equations indicate the normalizations chosen for these non-$L^2$ improper eigenfunctions of the differential operators (with derivatives respect to spatial co-ordinates) which generate them; the normalizations (1.17.12_1) and (1.17.12_2) are explicitly derived in Friedman (1990, Ch. 4), the others follow similarly. Equations (1.17.12_1) through (1.17.16) may re-interpreted as spectral representations of completeness relations, expressed in terms of Dirac delta distributions, as discussed in §1.18(v), and §1.18(vi) Further mathematical underpinnings are referenced in §1.17(iv). …

19: Bibliography M

… ►

Computation of inhomogeneous Airy functions. J. Comput. Appl. Math. 53 (1), pp. 109–116.

ⓘ

External Links:

ISSN 0377-0427, Document, MathReview (Dumitru Acu), ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

… ►

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

ⓘ

Notes:

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

External Links:

ISSN 0021-9991, Document, MathReview Entry, ZentralBlatt

Referenced by:

1st item.

Encodings:

BibTeX

… ►

Evaluation of complex logarithms and related functions. SIAM J. Numer. Anal. 18 (4), pp. 744–750.

ⓘ

External Links:

ISSN 0036-1429, Document, MathReview (M. Brannigan), ZentralBlatt

Referenced by:

§4.45(ii).

Encodings:

BibTeX

… ►

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

ⓘ

External Links:

MathReview (H. E. Fettis), ZentralBlatt

Referenced by:

§32.11(ii), §32.17, §32.3(ii).

Encodings:

BibTeX

… ►

A new symmetry related to $\mathrm{𝑆𝑈}(n)$ for classical basic hypergeometric series. Adv. in Math. 57 (1), pp. 71–90.

ⓘ

External Links:

ISSN 0001-8708, Document, MathReview (George E. Andrews), ZentralBlatt

Referenced by:

§17.15.

Encodings:

BibTeX

…

20: 18.15 Asymptotic Approximations

… ►

In Terms of Airy Functions

… ►Here $\mathrm{Ai}$ denotes the Airy function (§9.2), ${\mathrm{Ai}}^{\prime }$ denotes its derivative, and $\mathrm{envAi}$ denotes its envelope (§2.8(iii)). … ►And for asymptotic expansions as $n\to \mathrm{\infty }$ in terms of Airy functions that apply uniformly when or , see §§12.10(vii) and 12.10(viii). With $\mu =\sqrt{2n+1}$ the expansions in Chapter 12 are for the parabolic cylinder function $U(-\frac{1}{2}{\mu }^2,\mu t\sqrt{2})$, which is related to the Hermite polynomials via … ►For an error bound for the first term in the Airy-function expansions see Olver (1997b, p. 403). …