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11: 5.10 Continued Fractions

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$a_1=\frac{1}{30},$

… ►For exact values of $a_7$ to $a_{11}$ and 40S values of $a_0$ to $a_{40}$, see Char (1980). …

12: 24.20 Tables

… ►Wagstaff (1978) gives complete prime factorizations of $N_n$ and $E_n$ for $n=20(2)60$ and $n=8(2)42$, respectively. … ►For information on tables published before 1961 see Fletcher et al. (1962, v. 1, §4) and Lebedev and Fedorova (1960, Chapters 11 and 14).

13: Bibliography G

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Algorithm 292: Regular Coulomb wave functions. Comm. ACM 9 (11), pp. 793–795.

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External Links:

ISSN 0001-0782, Document

Referenced by:

§33.26(ii).

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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

External Links:

ISSN 0001-0782, Document

Referenced by:

§7.25(iii).

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

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External Links:

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

Referenced by:

§9.17(iii).

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Stirling number representation problems. Proc. Amer. Math. Soc. 11 (3), pp. 447–451.

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External Links:

FullText, MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§26.1, §26.1, Notations S, Notations S.

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Theory of Painlevé’s equations. Differ. Uravn. 11 (11), pp. 373–376 (Russian).

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Notes:

In Russian. English translation: Differential Equations 11(11), pp. 285–287

External Links:

MathReview (Z. Nehari), ZentralBlatt

Referenced by:

§32.7(iii), §32.7(vi).

Encodings:

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14: 25.20 Approximations

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Cody et al. (1971) gives rational approximations for $\zeta \left(s\right)$ in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are $0.5\le s\le 5$, $5\le s\le 11$, $11\le s\le 25$, $25\le s\le 55$. Precision is varied, with a maximum of 20S.

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15: Bibliography B

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Evaluation of the incomplete gamma function of imaginary argument by Chebyshev polynomials. Math. Comp. 15 (73), pp. 7–11.

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External Links:

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

Referenced by:

§8.7.

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Ramanujan’s theories of elliptic functions to alternative bases. Trans. Amer. Math. Soc. 347 (11), pp. 4163–4244.

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External Links:

ISSN 0002-9947, FullText, MathReview (J. Borwein), ZentralBlatt

Referenced by:

§15.8(v), §20.11(iii).

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Vortices in Ginzburg-Landau Equations. In Proceedings of the International Congress of Mathematicians, Vol. III (Berlin, 1998), pp. 11–19.

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External Links:

ISSN 1431-0643, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§17.17.

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Theoretical and experimental investigation of the elliptical annual ring antenna. IEEE Trans. Antennas and Propagation 36 (11), pp. 1526–1530.

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External Links:

Document

Referenced by:

5th item.

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Rainbow over Woolsthorpe Manor. Notes and Records Roy. Soc. London 36 (1), pp. 3–11 (1 plate).

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External Links:

ISSN 0035-9149, Document, MathReview

Referenced by:

Sidebar 9.SB1.

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16: Bibliography F

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Algorithm 838: Airy functions. ACM Trans. Math. Software 30 (4), pp. 491–501.

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Notes:

Single- and double-precision Fortran, maximum accuracy 32S.

External Links:

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

Referenced by:

3rd item, 2nd item, 1st item.

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Polynomial relations in the Heisenberg algebra. J. Math. Phys. 35 (11), pp. 6144–6149.

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External Links:

ISSN 0022-2488, Document, MathReview (Shao-Ming Fei), ZentralBlatt

Referenced by:

§13.3(ii), §15.5(i), §16.3(i).

Encodings:

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On a unified approach to transformations and elementary solutions of Painlevé equations. J. Math. Phys. 23 (11), pp. 2033–2042.

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External Links:

ISSN 0022-2488, Document, MathReview (Hélène Airault), ZentralBlatt

Referenced by:

§32.13(i), §32.7(i), §32.7(iii).

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The transformation properties of the sixth Painlevé equation and one-parameter families of solutions. Lett. Nuovo Cimento (2) 30 (17), pp. 539–544.

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External Links:

ISSN 0024-1318, MathReview (D. A. Lutz)

Referenced by:

§32.10(vi).

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Algorithm 435: Modified incomplete gamma function. Comm. ACM 15 (11), pp. 993–995.

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Notes:

Includes Fortran code. For a Remark see ACM Trans. Math. Software 4 (1978), no. 3, pp. 296–304.

External Links:

ISSN 0001-0782, Document, Remark

Referenced by:

§8.28(ii).

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17: 30 Spheroidal Wave Functions

Chapter 30 Spheroidal Wave Functions

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18: 10.75 Tables

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Zhang and Jin (1996, pp. 185–195) tabulates $J_n\left(x\right)$, $J_n^{\prime }\left(x\right)$, $Y_n\left(x\right)$, $Y_n^{\prime }\left(x\right)$, $n=0(1)10(10)50,100$, $x=1$, 5, 10, 25, 50, 100, 9S; $J_{n+\alpha }\left(x\right)$, $J_{n+\alpha }^{\prime }\left(x\right)$, $Y_{n+\alpha }\left(x\right)$, $Y_{n+\alpha }^{\prime }\left(x\right)$, $n=0(1)5,10,30,50,100$, $\alpha =\frac{1}{4},\frac{1}{3},\frac{1}{2},\frac{2}{3},\frac{3}{4}$, $x=1,5,10,50$, 8S; real and imaginary parts of $J_{n+\alpha }\left(z\right)$, $J_{n+\alpha }^{\prime }\left(z\right)$, $Y_{n+\alpha }\left(z\right)$, $Y_{n+\alpha }^{\prime }\left(z\right)$, $n=0(1)15,20(10)50,100$, $\alpha =0,\frac{1}{2}$, $z=4+2\mathrm{i}$, $20+10\mathrm{i}$, 8S.

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MacDonald (1989) tabulates the first 30 zeros, in ascending order of absolute value in the fourth quadrant, of the function $J_0\left(z\right)-\mathrm{i}{J}_1\left(z\right)$, 6D. (Other zeros of this function can be obtained by reflection in the imaginary axis).

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Abramowitz and Stegun (1964, Chapter 11) tabulates ${\int }_0^x{J}_0\left(t\right)dt$, ${\int }_0^x{Y}_0\left(t\right)dt$, $x=0(.1)10$, 10D; ${\int }_0^x{t}^{-1}(1-J_0\left(t\right))dt$, ${\int }_x^{\mathrm{\infty }}{t}^{-1}{Y}_0\left(t\right)dt$, $x=0(.1)5$, 8D.

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Zhang and Jin (1996, pp. 240–250) tabulates $I_n\left(x\right)$, $I_n^{\prime }\left(x\right)$, $K_n\left(x\right)$, $K_n^{\prime }\left(x\right)$, $n=0(1)10(10)50,100$, $x=1,5,10,25,50,100$, 9S; $I_{n+\alpha }\left(x\right)$, $I_{n+\alpha }^{\prime }\left(x\right)$, $K_{n+\alpha }\left(x\right)$, $K_{n+\alpha }^{\prime }\left(x\right)$, $n=0(1)5$, 10, 30, 50, 100, $\alpha =\frac{1}{4}$, $\frac{1}{3}$, $\frac{1}{2}$, $\frac{2}{3}$, $\frac{3}{4}$, $x=1$, 5, 10, 50, 8S; real and imaginary parts of $I_{n+\alpha }\left(z\right)$, $I_{n+\alpha }^{\prime }\left(z\right)$, $K_{n+\alpha }\left(z\right)$, $K_{n+\alpha }^{\prime }\left(z\right)$, $n=0(1)15$, 20(10)50, 100, $\alpha =0,\frac{1}{2}$, $z=4+2\mathrm{i},20+10\mathrm{i}$, 8S.

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Zhang and Jin (1996, pp. 296–305) tabulates ${𝗃}_n\left(x\right)$, ${𝗃}_n^{\prime }\left(x\right)$, ${𝗒}_n\left(x\right)$, ${𝗒}_n^{\prime }\left(x\right)$, ${𝗂}_n^{(1)}\left(x\right)$, $𝗂_n^{(1)}{}_{}{}^{\prime }\left(x\right)$, ${𝗄}_n\left(x\right)$, ${𝗄}_n^{\prime }\left(x\right)$, $n=0(1)10(10)30$, 50, 100, $x=1$, 5, 10, 25, 50, 100, 8S; $x{𝗃}_n\left(x\right)$, ${(x{𝗃}_n\left(x\right))}^{\prime }$, $x{𝗒}_n\left(x\right)$, ${(x{𝗒}_n\left(x\right))}^{\prime }$ (Riccati–Bessel functions and their derivatives), $n=0(1)10(10)30$, 50, 100, $x=1$, 5, 10, 25, 50, 100, 8S; real and imaginary parts of ${𝗃}_n\left(z\right)$, ${𝗃}_n^{\prime }\left(z\right)$, ${𝗒}_n\left(z\right)$, ${𝗒}_n^{\prime }\left(z\right)$, ${𝗂}_n^{(1)}\left(z\right)$, $𝗂_n^{(1)}{}_{}{}^{\prime }\left(z\right)$, ${𝗄}_n\left(z\right)$, ${𝗄}_n^{\prime }\left(z\right)$, $n=0(1)15$, 20(10)50, 100, $z=4+2\mathrm{i}$, $20+10\mathrm{i}$, 8S. (For the notation replace $j,y,\mathrm{i},k$ by $𝗃$, $𝗒$, ${𝗂}^{(1)}$, $𝗄$, respectively.)

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19: Bibliography K

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Bernstein, Pick, Poisson and related integral expressions for Lambert $W$ . Integral Transforms Spec. Funct. 23 (11), pp. 817–829.

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External Links:

ISSN 1065-2469, Document, Link, MathReview (Ross C. McPhedran), ZentralBlatt

Referenced by:

§4.13.

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Auxiliary table for the incomplete elliptic integrals. J. Math. Physics 27, pp. 11–36.

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External Links:

MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

§19.12.

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Gravitational field of a spinning mass as an example of algebraically special metrics. Phys. Rev. Lett. 11 (5), pp. 237–238.

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External Links:

Document, MathReview (C. Gilbert), ZentralBlatt

Referenced by:

§31.17(ii).

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Algorithm 327: Dilogarithm [S22]. Comm. ACM 11 (4), pp. 270–271.

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Notes:

Includes Algol program, maximum accuracy 12D.

External Links:

ISSN 0001-0782, Document

Referenced by:

§25.21(v).

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HYP and HYPQ. Mathematica packages for the manipulation of binomial sums and hypergeometric series respectively $q$-binomial sums and basic hypergeometric series. Séminaire Lotharingien de Combinatoire 30, pp. 61–76.

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Notes:

Also published in J. Symbolic Comput. (1995), v. 20, no. 5–6, pp. 737–744.

External Links:

Code, Document, MathReview (G. Gasper), ZentralBlatt

Referenced by:

1st item.

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20: 9.18 Tables

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Miller (1946) tabulates $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$ for $x=-20(.01)2;$ ${\log }_{10}\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)/\mathrm{Ai}\left(x\right)$ for $x=0(.1)25(1)75$; $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$ for $x=-10(.1)2.5$; ${\log }_{10}\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)/\mathrm{Bi}\left(x\right)$ for $x=0(.1)10$; $M\left(x\right)$, $N\left(x\right)$, $\theta \left(x\right)$, $\varphi \left(x\right)$ (respectively $F(x)$, $G(x)$, $\chi (x)$, $\psi (x)$) for $x=-80(1)-30(.1)0$. Precision is generally 8D; slightly less for some of the auxiliary functions. Extracts from these tables are included in Abramowitz and Stegun (1964, Chapter 10), together with some auxiliary functions for large arguments.

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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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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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National Bureau of Standards (1958) tabulates $A_0(x)\equiv \pi \mathrm{Hi}\left(-x\right)$ and $-A_0^{\prime }(x)\equiv \pi {\mathrm{Hi}}^{\prime }\left(-x\right)$ for $x=0(.01)1(.02)5(.05)11$ and $1/x=0.01(.01)0.1$; ${\int }_0^x{A}_0(t)dt$ for $x=0.5,1(1)11$. Precision is 8D.

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Gil et al. (2003c) tabulates the only positive zero of ${\mathrm{Gi}}^{\prime }\left(z\right)$, the first 10 negative real zeros of $\mathrm{Gi}\left(z\right)$ and ${\mathrm{Gi}}^{\prime }\left(z\right)$, and the first 10 complex zeros of $\mathrm{Gi}\left(z\right)$, ${\mathrm{Gi}}^{\prime }\left(z\right)$, $\mathrm{Hi}\left(z\right)$, and ${\mathrm{Hi}}^{\prime }\left(z\right)$. Precision is 11 or 12S.

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