relation to Airy function

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41—46 of 46 matching pages

41: Bibliography C

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Table of integrals of squared Jacobian elliptic functions and reductions of related hypergeometric $R$-functions. Math. Comp. 75 (255), pp. 1309–1318.

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External Links:

ISSN 0025-5718, Document, MathReview, ZentralBlatt

Referenced by:

§19.25(v), §19.29(i), §22.14(ii).

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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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External Links:

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

Referenced by:

§9.13(ii).

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Performance evaluation of programs related to the real gamma function. ACM Trans. Math. Software 17 (1), pp. 46–54.

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External Links:

ISSN 0098-3500, Document, MathReview, ZentralBlatt

Referenced by:

§5.24(ii), §5.24(iii).

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

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External Links:

ISSN 0048-6604, Document

Referenced by:

§9.14.

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Numerical evaluation of Airy functions with complex arguments. J. Comput. Phys. 99 (1), pp. 106–114.

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External Links:

ISSN 0021-9991, Document, MathReview, ZentralBlatt

Referenced by:

4th item, 1st item.

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42: 12.14 The Function $W(a,x)$

§12.14 The Function $W(a,x)$

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§12.14(vii) Relations to Other Functions

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Bessel Functions

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Confluent Hypergeometric Functions

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Airy-type Uniform Expansions

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43: 36.7 Zeros

… ►This is the Airy function $\mathrm{Ai}$ (§9.2). … ►Close to the $y$-axis the approximate location of these zeros is given by … ►Deep inside the bifurcation set, that is, inside the three-cusped astroid (36.4.10) and close to the part of the $z$-axis that is far from the origin, the zero contours form an array of rings close to the planes …Outside the bifurcation set (36.4.10), each rib is flanked by a series of zero lines in the form of curly “antelope horns” related to the “outside” zeros (36.7.2) of the cusp canonical integral. There are also three sets of zero lines in the plane $z=0$ related by $2\pi /3$ rotation; these are zeros of (36.2.20), whose asymptotic form in polar coordinates $(x=r\cos \theta ,y=r\sin \theta )$ is given by …

44: 12.10 Uniform Asymptotic Expansions for Large Parameter

… ►The turning points can be included if expansions in terms of Airy functions are used instead of elementary functions (§2.8(iii)). … ►

§12.10(vii) Negative $a$, . Expansions in Terms of Airy Functions

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Modified Expansions

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§12.10(viii) Negative $a$, . Expansions in Terms of Airy Functions

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45: Bibliography H

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Sums with cylindrical functions that reduce to the probability function and to related functions. Bul. Akad. Shtiintse RSS Moldoven. 1978 (3), pp. 80–84, 95 (Russian).

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External Links:

MathReview (M. Lawrence Glasser)

Referenced by:

§7.14(iii).

Encodings:

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An Introduction to the Theory of Numbers. 5th edition, The Clarendon Press Oxford University Press, New York-Oxford.

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External Links:

ISBN 0-19-853170-2; 0-19-853171-0, MathReview (T. M. Apostol), ZentralBlatt

Referenced by:

§20.12(i), Ch.27.

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On the distribution of the zeros of generalized Airy functions. Math. Comp. 29 (131), pp. 863–877.

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External Links:

FullText, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§9.13(i).

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An elementary approach to the hypergeometric shift operators of Opdam. Invent. Math. 103 (2), pp. 341–350.

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External Links:

ISSN 0020-9910, Document, MathReview (Eric M. Opdam), ZentralBlatt

Referenced by:

§18.37(iii).

Encodings:

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Introduction to Numerical Analysis. 2nd edition, McGraw-Hill Book Co., New York.

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Notes:

Reprinted by Dover Publications Inc., New York, 1987.

External Links:

ISBN 0-486-65363-3, MathReview, ZentralBlatt

Referenced by:

§3.3(iii), §3.3(iv), §3.3(v), §3.4(i), §3.8(v), Ch.3.

Encodings:

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46: 10.21 Zeros

… ►The functions ${\rho }_{\nu }(t)$ and ${\sigma }_{\nu }(t)$ are related to the inverses of the phase functions ${\theta }_{\nu }\left(x\right)$ and ${\varphi }_{\nu }\left(x\right)$ defined in §10.18(i): if $\nu \ge 0$, then … ►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. … ►Corresponding uniform approximations for $y_{\nu ,m}$, $Y_{\nu }^{\prime }\left(y_{\nu ,m}\right)$, $y_{\nu ,m}^{\prime }$, and $Y_{\nu }\left(y_{\nu ,m}^{\prime }\right)$, are obtained from (10.21.41)–(10.21.44) by changing the symbols $j$, $J$, $\mathrm{Ai}$, ${\mathrm{Ai}}^{\prime }$, $a_m$, and $a_m^{\prime }$ to $y$, $Y$, $-\mathrm{Bi}$, $-{\mathrm{Bi}}^{\prime }$, $b_m$, and $b_m^{\prime }$, respectively. … ►

§10.21(xiv) $\nu$-Zeros

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