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1: 9.18 Tables

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Zhang and Jin (1996, p. 337) tabulates $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$, $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$ for $x=0(1)20$ to 8S and for $x=-20(1)0$ to 9D.

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Miller (1946) tabulates $a_k$, ${\mathrm{Ai}}^{\prime }\left(a_k\right)$, $a_k^{\prime }$, $\mathrm{Ai}\left(a_k^{\prime }\right)$, $k=1(1)50$; $b_k$, ${\mathrm{Bi}}^{\prime }\left(b_k\right)$, $b_k^{\prime }$, $\mathrm{Bi}\left(b_k^{\prime }\right)$, $k=1(1)20$. Precision is 8D. Entries for $k=1(1)20$ are reproduced in Abramowitz and Stegun (1964, Chapter 10).

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Sherry (1959) tabulates $a_k$, ${\mathrm{Ai}}^{\prime }\left(a_k\right)$, $a_k^{\prime }$, $\mathrm{Ai}\left(a_k^{\prime }\right)$, $k=1(1)50$; 20S.

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Zhang and Jin (1996, p. 339) tabulates $a_k$, ${\mathrm{Ai}}^{\prime }\left(a_k\right)$, $a_k^{\prime }$, $\mathrm{Ai}\left(a_k^{\prime }\right)$, $b_k$, ${\mathrm{Bi}}^{\prime }\left(b_k\right)$, $b_k^{\prime }$, $\mathrm{Bi}\left(b_k^{\prime }\right)$, $k=1(1)20$; 8D.

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Smirnov (1960) tabulates $U_1(x,\alpha )$, $U_2(x,\alpha )$, defined by (9.13.20), (9.13.21), and also $\partial U_1(x,\alpha )/\partial x$, $\partial U_2(x,\alpha )/\partial x$, for $\alpha =1$, $x=-6(.01)10$ to 5D or 5S, and also for $\alpha =\pm \frac{1}{4}$, $\pm \frac{1}{3}$, $\pm \frac{1}{2}$, $\pm \frac{2}{3}$, $\pm \frac{3}{4}$, $\frac{5}{4}$, $\frac{4}{3}$, $\frac{3}{2}$, $\frac{5}{3}$, $\frac{7}{4}$, 2, $x=0(.01)6$; 4D.

2: 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(vi) Modifications of Expansions in Elementary Functions

… ►The proof of the double asymptotic property then follows with the aid of error bounds; compare §10.41(iv). … ►

Modified Expansions

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3: 10.75 Tables

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§10.75(ii) Bessel Functions and their Derivatives

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§10.75(iii) Zeros and Associated Values of the Bessel Functions, Hankel Functions, and their Derivatives

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Abramowitz and Stegun (1964, Chapter 9) tabulates $j_{n,m}$, $J_n^{\prime }\left(j_{n,m}\right)$, $j_{n,m}^{\prime }$, $J_n\left(j_{n,m}^{\prime }\right)$, $n=0(1)8$, $m=1(1)20$, 5D (10D for $n=0$), $y_{n,m}$, $Y_n^{\prime }\left(y_{n,m}\right)$, $y_{n,m}^{\prime }$, $Y_n\left(y_{n,m}^{\prime }\right)$, $n=0(1)8$, $m=1(1)20$, 5D (8D for $n=0$), $J_0\left(j_{0,m}x\right)$, $m=1(1)5$, $x=0(.02)1$, 5D. Also included are the first 5 zeros of the functions $x{J}_1\left(x\right)-\lambda J_0\left(x\right)$, $J_1\left(x\right)-\lambda x{J}_0\left(x\right)$, $J_0\left(x\right)Y_0\left(\lambda x\right)-Y_0\left(x\right)J_0\left(\lambda x\right)$, $J_1\left(x\right)Y_1\left(\lambda x\right)-Y_1\left(x\right)J_1\left(\lambda x\right)$, $J_1\left(x\right)Y_0\left(\lambda x\right)-Y_1\left(x\right)J_0\left(\lambda x\right)$ for various values of $\lambda$ and ${\lambda }^{-1}$ in the interval $[0,1]$, 4–8D.

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§10.75(ix) Spherical Bessel Functions, Modified Spherical Bessel Functions, and their Derivatives

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§10.75(xii) Zeros of Kelvin Functions and their Derivatives

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4: Bibliography B

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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.

External Links:

Document, Code, ZentralBlatt

Referenced by:

1st item.

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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.

External Links:

Document, Code

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1st item, 4th item.

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Overlapping Stokes smoothings: Survival of the error function and canonical catastrophe integrals. Proc. Roy. Soc. London Ser. A 444, pp. 201–216.

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External Links:

ISSN 0962-8444, FullText, MathReview (F. Ursell), ZentralBlatt

Referenced by:

§36.12(iii).

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Rational Chebyshev approximations for the inverse of the error function. Math. Comp. 30 (136), pp. 827–830.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§7.17(iii), §7.17(iii).

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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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5: Bibliography S

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Transformations of the Jacobian amplitude function and its calculation via the arithmetic-geometric mean. SIAM J. Math. Anal. 20 (6), pp. 1514–1528.

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External Links:

ISSN 0036-1410, Document, MathReview (J. M. H. Peters), ZentralBlatt

Referenced by:

§22.19(i), §22.20(vi).

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Chebyshev expansions for the error and related functions. Math. Comp. 32 (144), pp. 1232–1240.

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External Links:

FullText, Document, MathReview (L. Fox), ZentralBlatt

Referenced by:

3rd item.

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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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A Maple package for symmetric functions. J. Symbolic Comput. 20 (5-6), pp. 755–768.

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Notes:

For software see John Stembridge’s home page. The SF package contains Maple software for zonal, Jack, and Macdonald polynomials.

External Links:

Home Page, ISSN 0747-7171, Document, MathReview, ZentralBlatt

Referenced by:

3rd item.

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On the calculation of the inverse of the error function. Math. Comp. 22 (101), pp. 144–158.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

§7.17(ii).

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6: Bibliography G

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Algorithm 363: Complex error function. Comm. ACM 12 (11), pp. 635.

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Notes:

Includes Algol program, maximum accuracy 10S. For certification see same journal v. 15 (1972), pp. 465–466

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ISSN 0001-0782, Document

Referenced by:

§7.25(iii).

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Recursive computation of the repeated integrals of the error function. Math. Comp. 15 (75), pp. 227–232.

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External Links:

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

Referenced by:

§7.22(iii).

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Efficient computation of the complex error function. SIAM J. Numer. Anal. 7 (1), pp. 187–198.

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External Links:

Document, MathReview (P. Barrucand), ZentralBlatt

Referenced by:

§7.22(i), §7.25(iii).

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A table of integrals of the error function. II. Additions and corrections. J. Res. Nat. Bur. Standards Sect. B 75B, pp. 149–163.

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Notes:

See for the first part same journal, Sect. B 73, 1-20 (1969)

External Links:

MathReview, ZentralBlatt

Referenced by:

§7.14(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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7: Bibliography C

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The inverse of the error function. Pacific J. Math. 13 (2), pp. 459–470.

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External Links:

MathReview (D. P. Banerjee), ZentralBlatt

Referenced by:

§7.17(ii).

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The Riemann zeta-function and its derivatives. Proc. Roy. Soc. London Ser. A 450, pp. 477–499.

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External Links:

ISSN 0962-8444, FullText, MathReview, ZentralBlatt

Referenced by:

§25.18(i).

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Rational Chebyshev approximations for the error function. Math. Comp. 23 (107), pp. 631–637.

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External Links:

FullText, MathReview, ZentralBlatt

Referenced by:

2nd item.

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

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External Links:

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

Referenced by:

§13.29(v), §15.19(v).

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Zeros of the Hankel function of real order and of its derivative. Math. Comp. 39 (160), pp. 639–645.

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External Links:

ISSN 0025-5718, FullText, MathReview (S. Ahmed), ZentralBlatt

Referenced by:

§10.21(ix).

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8: Bibliography V

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A Fortran computer program for calculating the oblate spheroidal radial functions of the first and second kind and their first derivatives. NRL Report No. 6959 Naval Res. Lab.  Washingtion, D.C..

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Referenced by:

§30.18(iii).

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Accurate calculation of prolate spheroidal radial functions of the first kind and their first derivatives. Quart. Appl. Math. 60 (3), pp. 589–599.

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Notes:

For software, see Van Buren (website)

External Links:

ISSN 0033-569X, MathReview (Alan M. Cohen), ZentralBlatt

Referenced by:

§30.16(iii).

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Improved calculation of prolate spheroidal radial functions of the second kind and their first derivatives. Quart. Appl. Math. 62 (3), pp. 493–507.

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Notes:

For software, see Van Buren (website)

External Links:

ISSN 0033-569X, FullText, MathReview (Javier Segura), ZentralBlatt

Referenced by:

§30.16(iii), §30.16(iii).

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Calculation of Special Functions: The Gamma Function, the Exponential Integrals and Error-Like Functions. CWI Tract, Vol. 10, Stichting Mathematisch Centrum, Centrum voor Wiskunde en Informatica, Amsterdam.

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Notes:

A reprint was published in 1986.

External Links:

ISBN 90-6196-277-3, MathReview (John P. Coleman), ZentralBlatt

Referenced by:

§5.21, §6.18(iv), §7.22(v), §7.6(i), §7.7(i), §7.7(ii).

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Error estimates for Rayleigh-Ritz approximations of eigenvalues and eigenfunctions of the Mathieu and spheroidal wave equation. Constr. Approx. 20 (1), pp. 39–54.

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External Links:

ISSN 0176-4276, Document, MathReview, ZentralBlatt

Referenced by:

§30.16(i), §30.16(ii), §30.16(ii).

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9: 3.8 Nonlinear Equations

… ►For other efficient derivative-free methods, see Le (1985). … ►However, to guard against the accumulation of rounding errors, a final iteration for each zero should also be performed on the original polynomial $p(z)$. … ►

§3.8(v) Zeros of Analytic Functions

… ►Consider $x=20$ and $j=19$. We have $p^{\prime }(20)=19!$ and $a_{19}=1+2+\mathrm{\cdots }+20=210$. …

10: Bibliography L

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Monotonicity results and inequalities for the gamma and error functions. J. Comput. Appl. Math. 23 (1), pp. 25–33.

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External Links:

ISSN 0377-0427, Document, MathReview (M. E. Cohen), ZentralBlatt

Referenced by:

§7.8.

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On the Zeros of the Derivative of Bessel Functions of Second Kind. Pubblicazioni Serie III [Publication Series III], Vol. 179, Istituto per le Applicazioni del Calcolo “Mauro Picone” (IAC), Rome.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§10.21(vi).

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Algorithm 917: complex double-precision evaluation of the Wright $\omega$ function. ACM Trans. Math. Software 38 (3), pp. Art. 20, 17.

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External Links:

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

Referenced by:

§4.13, 2nd item, §4.48(iv).

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Monotonicity of the zeros of the third derivative of Bessel functions. Methods Appl. Anal. 2 (1), pp. 103–111.

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External Links:

ISSN 1073-2772, FullText, MathReview (N. Hayek Calil), ZentralBlatt

Referenced by:

§10.21(iv).

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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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External Links:

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

Referenced by:

§10.21(iv), §10.21(iv).

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