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1: 30.9 Asymptotic Approximations and Expansions

… ►For uniform asymptotic expansions in terms of Airy or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1986). … ►

§30.9(ii) Oblate Spheroidal Wave Functions

… ►For uniform asymptotic expansions in terms of elementary, Airy, or Bessel functions for real values of the parameters, complex values of the variable, and with explicit error bounds see Dunster (1992, 1995). … ►

§30.9(iii) Other Approximations and Expansions

►The asymptotic behavior of ${\lambda }_n^m\left({\gamma }^2\right)$ and $a_{n,k}^m({\gamma }^2)$ as $n\to \mathrm{\infty }$ in descending powers of $2n+1$ is derived in Meixner (1944). …

2: 36.5 Stokes Sets

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$K=1$. Airy Function

… ►For $z\ne 0$, the Stokes set is expressed in terms of scaled coordinates …in which … ►in which … ►The distribution of real and complex critical points in Figures 36.5.5 and 36.5.6 follows from consistency with Figure 36.5.1 and the fact that there are four real saddles in the inner regions. …

3: 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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4: 12.11 Zeros

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§12.11(i) Distribution of Real Zeros

… ►For large negative values of $a$ the real zeros of $U(a,x)$, $U^{\prime }(a,x)$, $V(a,x)$, and $V^{\prime }(a,x)$ can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). …Here $\alpha ={\mu }^{-\frac{4}{3}}{a}_s$, $a_s$ denoting the $s$th negative zero of the function $\mathrm{Ai}$ (see §9.9(i)). … ►where $\beta ={\mu }^{-\frac{4}{3}}{a}_s^{\prime }$, $a_s^{\prime }$ denoting the $s$th negative zero of the function ${\mathrm{Ai}}^{\prime }$ and … ►For further information, including associated functions, see Olver (1959).

5: 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).

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A program for computing the Riemann zeta function for complex argument. Comput. Phys. Comm. 20 (3), pp. 441–445.

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Notes:

Double-precision Fortran, maximum accuracy 10S.

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Document, Code, ZentralBlatt

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1st item.

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The Bounds for the Error Term of an Asymptotic Approximation of Jacobi Polynomials. In Orthogonal Polynomials and Their Applications (Segovia, 1986), Lecture Notes in Math., Vol. 1329, pp. 203–221.

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External Links:

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

Referenced by:

§18.15(i), §18.15(i).

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Coulomb functions (negative energies). Comput. Phys. Comm. 20 (3), pp. 447–458.

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Notes:

Double-precision Fortran code.

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Document, Code

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1st item, 4th item.

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Bessel functions and modular relations of higher type and hyperbolic differential equations. Comm. Sém. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.] 1952 (Tome Supplementaire), pp. 12–20.

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External Links:

MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§35.5(i).

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6: Bibliography S

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Uniform asymptotic forms of modified Mathieu functions. Quart. J. Mech. Appl. Math. 20 (3), pp. 365–380.

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External Links:

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

Referenced by:

§28.8(iv).

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

Referenced by:

§2.5(ii).

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Numerical Methods Based on Sinc and Analytic Functions. Springer Series in Computational Mathematics, Vol. 20, Springer-Verlag, New York.

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Notes:

Sinc-Pack, a set of 480 Matlab programs with a 470-page tutorial, is available for purchase from SINC, LLC by contacting Frank Stenger.

External Links:

ISBN 0-387-94008-1, MathReview (B. Boyanov), ZentralBlatt

Referenced by:

§3.3(vi), §3.4(i), §3.5(i).

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7: Bibliography G

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

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

Referenced by:

§10.74(iii), §9.17(iii).

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

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Notes:

Problem proposed by P. Smith.

External Links:

Document

Referenced by:

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

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On nonoscillating integrals for computing inhomogeneous Airy functions. Math. Comp. 70 (235), pp. 1183–1194.

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Document, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§3.5(ix), §9.17(iii).

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Computing complex Airy functions by numerical quadrature. Numer. Algorithms 30 (1), pp. 11–23.

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ISSN 1017-1398, Document, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§9.17(iii).

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Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:

Includes Fortran program claiming 12S accuracy

External Links:

ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

3rd item, §10.77(ix).

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8: 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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Validated computation of certain hypergeometric functions. ACM Trans. Math. Software 38 (2), pp. Art. 11, 20.

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ISSN 0098-3500, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§13.29(v), §15.19(v).

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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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Coulomb effects in the Klein-Gordon equation for pions. Phys. Rev. C 20 (2), pp. 696–704.

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Document

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5th item.

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9: Bibliography

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Regular and irregular Coulomb wave functions expressed in terms of Bessel-Clifford functions. J. Math. Physics 33, pp. 111–116.

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External Links:

MathReview (E. Weber), ZentralBlatt

Referenced by:

§33.9(ii).

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On the intensity of light in the neighbourhood of a caustic. Trans. Camb. Phil. Soc. 6, pp. 379–402.

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Referenced by:

§9.16, §9.16.

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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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Repeated integrals and derivatives of $K$ Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.

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ISSN 0036-1410, Document, MathReview (A. Haimovici), ZentralBlatt

Referenced by:

§10.43(iii).

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10: Bibliography M

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

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

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1st 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 $x$.

External Links:

ISSN 0021-9991, Document, MathReview Entry, ZentralBlatt

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1st item.

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Hierarchies and logarithmic oscillations in the temporal relaxation patterns of proteins and other complex systems. Proc. Nat. Acad. Sci. U .S. A. 96 (20), pp. 11085–11089.

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FullText

Referenced by:

§8.24(i).

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

Referenced by:

§29.11, §29.7(ii).

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