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21: 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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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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Von Staudt for ${𝐅}_q[T]$ . Duke Math. J. 45 (4), pp. 885–910.

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ISSN 0012-7094, Document, MathReview (Neal Koblitz), ZentralBlatt

Referenced by:

§24.16(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).

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22: Bibliography B

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A monotonicity property involving ${}_3{}^{}F_2^{}$ and comparisons of the classical approximations of elliptical arc length. SIAM J. Math. Anal. 32 (2), pp. 403–419.

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External Links:

ISSN 1095-7154, Document, MathReview (Richard B. Paris), ZentralBlatt

Referenced by:

§19.9(i), §19.9(i).

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Zero-field susceptibility of the two-dimensional Ising model near $T_c$ . Phys. Rev. Lett. 31, pp. 1409–1411.

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External Links:

Document

Referenced by:

§32.16.

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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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ISSN 1431-0643, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§17.17.

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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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ISSN 0035-9149, Document, MathReview

Referenced by:

Sidebar 9.SB1.

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The behavior at unit argument of the hypergeometric function ${}_3{}^{}F_2^{}$ . SIAM J. Math. Anal. 18 (5), pp. 1227–1234.

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External Links:

ISSN 0036-1410, Document, MathReview (K. M. Saksena), ZentralBlatt

Referenced by:

§16.8(ii).

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23: Bibliography F

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Tables of Values of the Function $w(z)=e^{-z^2}(1+2i{\pi }^{-1/2}{\int }_0^z{e}^{t^2}𝑑t)$ for Complex Argument. Edited by V. A. Fok; translated from the Russian by D. G. Fry. Mathematical Tables Series, Vol. 11, Pergamon Press, Oxford.

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External Links:

MathReview, ZentralBlatt

Referenced by:

V. N. Faddeeva and N. M. Terent’ev (1954).

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

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

Referenced by:

§13.3(ii), §15.5(i), §16.3(i).

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Application of the $\tau$-function theory of Painlevé equations to random matrices: ${\mathrm{P}}_{\mathrm{V}}$, ${\mathrm{P}}_{\mathrm{III}}$, the LUE, JUE, and CUE. Comm. Pure Appl. Math. 55 (6), pp. 679–727.

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External Links:

ISSN 0010-3640, Document, MathReview (Mark Adler), ZentralBlatt

Referenced by:

§32.10(iii), §32.10(v), §32.14, §32.6(iv), §32.6(v).

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Application of the $\tau$-function theory of Painlevé equations to random matrices: ${\mathrm{P}}_{\mathrm{VI}}$, the JUE, CyUE, cJUE and scaled limits. Nagoya Math. J. 174, pp. 29–114.

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ISSN 0027-7630, MathReview, ZentralBlatt

Referenced by:

§32.10(vi), §32.6(v).

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Error bounds for a uniform asymptotic expansion of the Legendre function $Q_n^{-m}(\cosh z)$ . SIAM J. Math. Anal. 21 (2), pp. 523–535.

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External Links:

ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§14.26.

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24: 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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Harvard University (1945) tabulates the real and imaginary parts of $h_1(z)$, $h_1^{\prime }(z)$, $h_2(z)$, $h_2^{\prime }(z)$ for $-x_0\le \mathrm{\Re }z\le x_0$, $0\le \mathrm{\Im }z\le y_0$, , with interval 0.1 in $\mathrm{\Re }z$ and $\mathrm{\Im }z$. Precision is 8D. Here $h_1(z)=-2^{4/3}{3}^{1/6}\mathrm{i}\mathrm{Ai}\left({\mathrm{e}}^{-\pi \mathrm{i}/3}z\right)$, $h_2(z)=2^{4/3}{3}^{1/6}\mathrm{i}\mathrm{Ai}\left({\mathrm{e}}^{\pi \mathrm{i}/3}z\right)$.

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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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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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25: 18.8 Differential Equations

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Table 18.8.1: Classical OP’s: differential equations $A(x)f^{\prime \prime }(x)+B(x)f^{\prime }(x)+C(x)f(x)+{\lambda }_{n}f(x)=0$.

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#	$f(x)$	$A(x)$	$B(x)$	$C(x)$	${\lambda }_n$
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11	${\mathrm{e}}^{-n^{-1}x}{x}^{\mathrm{\ell }+1}{L}_{n-\mathrm{\ell }-1}^{(2\mathrm{\ell }+1)}\left(2n^{-1}x\right)$	$1$	$0$	$\frac{2}{x}-\frac{\mathrm{\ell }(\mathrm{\ell }+1)}{x^2}$	$-\frac{1}{n^2}$
12	$H_n\left(x\right)$	$1$	$-2x$	$0$	$2n$
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14	${\mathrm{𝐻𝑒}}_n\left(x\right)$	$1$	$-x$	$0$	$n$

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►Item 11 of Table 18.8.1 yields (18.39.36) for $Z=1$.

26: 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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Morris (1979) gives rational approximations for ${\mathrm{Li}}_2\left(x\right)$ (§25.12(i)) for $0.5\le x\le 1$. Precision is varied with a maximum of 24S.

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Antia (1993) gives minimax rational approximations for $\mathrm{\Gamma }\left(s+1\right)F_s(x)$, where $F_s(x)$ is the Fermi–Dirac integral (25.12.14), for the intervals and , with $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$. For each $s$ there are three sets of approximations, with relative maximum errors ${10}^{-4},{10}^{-8},{10}^{-12}$.

27: 21.1 Special Notation

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$g,h$	positive integers.
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$a_j$	$j$th element of vector $𝐚$.
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$\mathrm{diag}𝐀$	Transpose of $[A_{11},A_{22},\mathrm{\dots },A_{gg}]$.
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${𝐉}_{2g}$	.
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$S_1{S}_2$	set of all elements of the form “$\text{element of }{S}_1\times \text{element of }{S}_2$”.
$S_1/S_2$	set of all elements of $S_1$, modulo elements of $S_2$. Thus two elements of $S_1/S_2$ are equivalent if they are both in $S_1$ and their difference is in $S_2$. (For an example see §20.12(ii).)
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28: Bibliography L

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Ratios of Bessel functions and roots of $\alpha J_{\nu }(x)+x{J}_{\nu }^{\prime }(x)=0$ . J. Math. Anal. Appl. 240 (1), pp. 174–204.

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External Links:

ISSN 0022-247X, Document, MathReview (Boro Döring), ZentralBlatt

Referenced by:

§10.14, §10.15, §10.21(xii).

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A table of integrals involving the functions $E_n(x)$ .

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Notes:

National Research Council of Canada, Division of Atomic Energy, Document no. MT-131 (NRC 1553)

Referenced by:

§8.19(x).

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Eine Verallgemeinerung der Sphäroidfunktionen. Arch. Math. 11, pp. 29–39.

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ISSN 0003-9268, Document, MathReview (M. J. O. Strutt), ZentralBlatt

Referenced by:

§30.12.

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Note sur la fonction $𝔎(w,x,s)={\sum }_{k=0}^{\mathrm{\infty }}\frac{e^{2k\pi ix}}{{(w+k)}^s}$ . Acta Math. 11 (1-4), pp. 19–24 (French).

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External Links:

ISSN 0001-5962, Document, MathReview Entry, ZentralBlatt

Referenced by:

§25.14(i), Notations K.

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On the theory of Painlevé’s third equation. Differ. Uravn. 3 (11), pp. 1913–1923 (Russian).

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Notes:

In Russian. English translation: Differential Equations 3(11), pp. 994–999

External Links:

MathReview (Z. Nehari), ZentralBlatt

Referenced by:

§32.10(iii), §32.8(iii), §32.9(i), §32.9(ii).

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29: Bibliography S

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Note on the complex zeros of $H_{\nu }^{\prime }(x)+i\zeta H_{\nu }(x)=0$ . J. Comput. Appl. Math. 201 (1), pp. 3–7.

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External Links:

ISSN 0377-0427, Document, FullText, MathReview, ZentralBlatt

Referenced by:

§10.21(xiv), §10.21(xiv).

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Recursive evaluation of $3j$- and $6j$- coefficients. Comput. Phys. Comm. 11 (2), pp. 269–278.

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Notes:

Double-precision Fortran provided in 3 parts, $j_1$-recursion for $3j$-coefficients, $m_2$-recursion for $3j$-coefficients, and $j_1$-recursion for $6j$-coefficients

External Links:

Document, Code 1, Code 2, Code 3

Referenced by:

8th item.

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Numerical evaluation of integrals of the form $I={\int }_{x_1}^{x_2}f(x)e^{i\varphi (x)}𝑑x$ and the tabulation of the function $\mathrm{Gi}(z)=(1/\pi ){\int }_0^{\mathrm{\infty }}\sin (uz+\frac{1}{3}{u}^3)𝑑u$ . Quart. J. Mech. Appl. Math. 3 (1), pp. 107–112.

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External Links:

Document, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

1st item.

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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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On the roots of $f(z)=J_0(z)-i{J}_1(z)$ . Quart. Appl. Math. 46 (1), pp. 105–107.

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External Links:

ISSN 0033-569X, MathReview (Evangelos Ifantis), ZentralBlatt

Referenced by:

§10.21(ix).

Encodings:

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30: 26.4 Lattice Paths: Multinomial Coefficients and Set Partitions

… ► $\left(\genfrac{}{}{0.0pt}{}{n}{n_1,n_2,\mathrm{\dots },n_k}\right)$ is the number of ways of placing $n=n_1+n_2+\mathrm{\cdots }+n_k$ distinct objects into $k$ labeled boxes so that there are $n_j$ objects in the $j$th box. … ►These are given by the following equations in which $a_1,a_2,\mathrm{\dots },a_n$ are nonnegative integers such that …$M_1$ is the multinominal coefficient (26.4.2): …For each $n$ all possible values of $a_1,a_2,\mathrm{\dots },a_n$ are covered. … ►where the summation is over all nonnegative integers $n_1,n_2,\mathrm{\dots },n_k$ such that $n_1+n_2+\mathrm{\cdots }+n_k=n$. …