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11: Bibliography

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Some orthogonal $q$-polynomials. Math. Nachr. 30, pp. 47–61.

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ISSN 0025-584X, Document, MathReview (A. E. Danese), ZentralBlatt

Referenced by:

§18.27(vii).

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Numerical Calculation of the Riemann Zeta Function and Generalizations by Means of the Trapezoidal Rule. In Numerical and Applied Mathematics, Part II (Paris, 1988), C. Brezinski (Ed.), IMACS Ann. Comput. Appl. Math., Vol. 1, pp. 467–472.

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MathReview

Referenced by:

§25.18(i).

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A proof that Euler missed: Evaluating $\zeta (2)$ the easy way. Math. Intelligencer 5 (3), pp. 59–60.

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ISSN 0343-6993, MathReview (P. Szüsz), ZentralBlatt

Referenced by:

(25.6.7).

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Critical points of smooth functions, and their normal forms. Uspehi Mat. Nauk 30 (5(185)), pp. 3–65 (Russian).

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Notes:

In Russian. English translation: Russian Math. Surveys, 30(1975), no. 5, pp. 1–75.

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Document, MathReview (D. Siersma), ZentralBlatt

Referenced by:

§36.2(i), Ch.36.

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Mathematical Methods of Classical Mechanics. Graduate Texts in Mathematics, Vol. 60, Springer-Verlag, New York.

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Translated from the 1974 Russian original by K. Vogtmann and A. Weinstein, Corrected reprint of the second (1989) edition

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ISBN 0-387-96890-3, MathReview

Referenced by:

§21.5(i).

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12: Bibliography S

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Ein Verfahren zur Berechnung des charakteristischen Exponenten der Mathieuschen Differentialgleichung I. Numer. Math. 3 (1), pp. 30–38.

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ISSN 0029-599X, Document, MathReview (S. C. van Veen), ZentralBlatt

Referenced by:

item (b).

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Sharper bounds for the Chebyshev functions $\theta (x)$ and $\psi (x)$. II. Math. Comp. 30 (134), pp. 337–360.

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Notes:

For corrigendum see same journal v. 30 (1976), no. 136, p. 900

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FullText, Document, MathReview (R. P. Brent), ZentralBlatt

Referenced by:

§27.12.

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An algorithm for the evaluation of the complex Airy functions. J. Comput. Phys. 31 (1), pp. 60–75.

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ISSN 0021-9991, Document, MathReview, ZentralBlatt

Referenced by:

(9.10.18), (9.10.19), §9.17(iii), Erratum (V1.0.10) for Equation (9.10.18), Erratum (V1.0.10) for Equation (9.10.19).

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Confluent Hypergeometric Functions. Cambridge University Press, Cambridge-New York.

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Table errata: Math. Comp. v. 30 (1976), no. 135, 677–678

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MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§13.1, §13.10(ii), §13.10(iii), (13.11.3), §13.11, §13.13(i), §13.13(ii), §13.13(iii), §13.14(i), §13.14(vi), §13.15(i), §13.15(ii), §13.16(ii), §13.16(iii), §13.2(v), §13.2(vi), §13.21(i), §13.23(i), §13.24, §13.26(i), §13.26(ii), §13.26(iii), §13.3(i), §13.3(ii), 2nd item, §13.4(i), §13.4(ii), §13.8(i), §13.9(i), §13.9(ii), Ch.13, §8.5.

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Structure of avoided crossings for eigenvalues related to equations of Heun’s class. J. Phys. A 30 (2), pp. 673–687.

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ISSN 0305-4470, Document, MathReview (Éric Delabaere), ZentralBlatt

Referenced by:

§31.13.

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13: Bibliography F

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Computation of complex Airy functions and their zeros using asymptotics and the differential equation. ACM Trans. Math. Software 30 (4), pp. 471–490.

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ISSN 0098-3500, Document, MathReview, ZentralBlatt

Referenced by:

§9.17(ii), §9.17(v), §9.8(iv).

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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.

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

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

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The third Appell function for one large variable. J. Approx. Theory 165, pp. 60–69.

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ISSN 0021-9045, Document, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§16.13, §16.13.

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Complex roots of $\sin z=az$, $\cos z=az$, and $\cosh z=az$ . Math. Comp. 30 (135), pp. 541–545.

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FullText, MathReview (Gerhard Merz), ZentralBlatt

Referenced by:

§4.46.

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

Referenced by:

§32.10(vi).

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14: 29.21 Tables

§29.21 Tables

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Ince (1940a) tabulates the eigenvalues $a_{\nu }^m\left(k^2\right)$, $b_{\nu }^{m+1}\left(k^2\right)$ (with $a_{\nu }^{2m+1}$ and $b_{\nu }^{2m+1}$ interchanged) for $k^2=0.1,0.5,0.9$, $\nu =-\frac{1}{2},0(1)25$, and $m=0,1,2,3$. Precision is 4D.

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Arscott and Khabaza (1962) tabulates the coefficients of the polynomials $P$ in Table 29.12.1 (normalized so that the numerically largest coefficient is unity, i.e. monic polynomials), and the corresponding eigenvalues $h$ for $k^2=0.1(.1)0.9$, $n=1(1)30$. Equations from §29.6 can be used to transform to the normalization adopted in this chapter. Precision is 6S.

15: Bibliography L

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Application of theta functions for numerical evaluation of complete elliptic integrals of the first and second kinds. Comput. Phys. Comm. 60 (3), pp. 319–327.

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ISSN 0010-4655, Document, MathReview, ZentralBlatt

Referenced by:

§19.36(iii), §19.5.

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On the Darling-Mandelbrot probability density and the zeros of some incomplete gamma functions. Constr. Approx. 10 (1), pp. 15–30.

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ISSN 0176-4276, Document, MathReview (P. Anandani), ZentralBlatt

Referenced by:

§8.13(i).

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Miniaturized tables of Bessel functions. Math. Comp. 25 (114), pp. 323–330.

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ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

§10.76(ii).

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Miniaturized tables of Bessel functions. III. Math. Comp. 26 (117), pp. 237–240 and A14–B5.

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ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

§10.76(ii).

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Highly accurate tables for elementary functions. BIT 35 (3), pp. 352–360.

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ISSN 0006-3835, Document, MathReview, ZentralBlatt

Referenced by:

§4.45(i), §4.46.

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16: 25.19 Tables

§25.19 Tables

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Abramowitz and Stegun (1964) tabulates: $\zeta \left(n\right)$, $n=2,3,4,\mathrm{\dots }$, 20D (p. 811); ${\mathrm{Li}}_2\left(1-x\right)$, $x=0(.01)0.5$, 9D (p. 1005); $f(\theta )$, $\theta ={15}^{\circ }(1^{\circ }){30}^{\circ }(2^{\circ }){90}^{\circ }(5^{\circ }){180}^{\circ }$, $f(\theta )+\theta \ln \theta$, $\theta =0(1^{\circ }){15}^{\circ }$, 6D (p. 1006). Here $f(\theta )$ denotes Clausen’s integral, given by the right-hand side of (25.12.9).

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Morris (1979) tabulates ${\mathrm{Li}}_2\left(x\right)$ (§25.12(i)) for $\pm x=0.02(.02)1(.1)6$ to 30D.

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Cloutman (1989) tabulates $\mathrm{\Gamma }\left(s+1\right)F_s(x)$, where $F_s(x)$ is the Fermi–Dirac integral (25.12.14), for $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$, $x=-5(.05)25$, to 12S.

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Fletcher et al. (1962, §22.1) lists many sources for earlier tables of $\zeta \left(s\right)$ for both real and complex $s$. §22.133 gives sources for numerical values of coefficients in the Riemann–Siegel formula, §22.15 describes tables of values of $\zeta (s,a)$, and §22.17 lists tables for some Dirichlet $L$-functions for real characters. For tables of dilogarithms, polylogarithms, and Clausen’s integral see §§22.84–22.858.

17: Bibliography M

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Chebyshev expansions for modified Struve and related functions. Math. Comp. 60 (202), pp. 735–747.

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ISSN 0025-5718, FullText, MathReview, ZentralBlatt

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2nd item.

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Formulas and Theorems for the Special Functions of Mathematical Physics. 3rd edition, Springer-Verlag, New York-Berlin.

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Notes:

Table errata: Math. Comp. v. 23 (1969), p. 471, v. 24 (1970), pp. 240 and 505, v. 25 (1971), no. 113, p. 201, v. 29 (1975), no. 130, p. 672, v. 30 (1976), no. 135, pp. 677–678, v. 32 (1978), no. 141, pp. 319–320, v. 36 (1981), no. 153, pp. 315–317, v. 41 (1983), no. 164, pp. 775–776, and v. 81 (2012), no. 280, p. 2251

External Links:

MathReview (Mary L. Boas), ZentralBlatt

Referenced by:

§10.15, §10.22(vi), §10.32(iv), §10.38, §10.43(vi), §10.60(iv), §10.9(iv), §11.5(iii), Ch.11, §12.12, §12.5(iv), Ch.12, §13.10(vi), §13.23(iv), §14.1, §14.11, §14.12(ii), §14.18(iv), §14.25, §14.3(iii), §14.5(v), §14.7(iv), Ch.14, (25.5.21), §25.5, (25.6.1), (25.6.2), §25.6(i), §8.1, Notations B, Notations I, Notations P, Notations P, Notations Q, Notations Q.

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On the roots of the Bessel and certain related functions. Ann. of Math. 9 (1-6), pp. 23–30.

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ISSN 0003-486X, FullText, MathReview Entry, ZentralBlatt

Referenced by:

§10.21(x).

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A $q$-analog of hypergeometric series well-poised in $\mathrm{𝑆𝑈}(n)$ and invariant $G$-functions. Adv. in Math. 58 (1), pp. 1–60.

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ISSN 0001-8708, Document, MathReview (Robert A. Gustafson), ZentralBlatt

Referenced by:

§17.15.

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New sharp bounds for gamma and digamma functions. An. Ştiinţ. Univ. Al. I. Cuza Iaşi. Mat. (N.S.) 57 (1), pp. 57–60.

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ISSN 1221-8421, Document, MathReview (Daniele Ritelli), ZentralBlatt

Referenced by:

§5.6(i).

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18: 30 Spheroidal Wave Functions

Chapter 30 Spheroidal Wave Functions

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19: 6.19 Tables

§6.19 Tables

… ►Lebedev and Fedorova (1960) and Fletcher et al. (1962) give comprehensive indexes of mathematical tables. This section lists relevant tables that appeared later. … ►

Zhang and Jin (1996, pp. 652, 689) includes $\mathrm{Si}\left(x\right)$, $\mathrm{Ci}\left(x\right)$, $x=0(.5)20(2)30$, 8D; $\mathrm{Ei}\left(x\right)$, $E_1\left(x\right)$, $x=[0,100]$, 8S.

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Zhang and Jin (1996, pp. 690–692) includes the real and imaginary parts of $E_1\left(z\right)$, $\pm x=0.5,1,3,5,10,15,20,50,100$, $y=0(.5)1(1)5(5)30,50,100$, 8S.

20: Bibliography Y

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On rational solutions of the second Painlevé equation. Vesti Akad. Navuk. BSSR Ser. Fiz. Tkh. Nauk. 3, pp. 30–35 (Russian).

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Referenced by:

§32.8(ii).

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Tables of Airy Functions and Their Derivatives. Izdat. Nauka, Moscow (Russian).

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Referenced by:

4th item.

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Partitions with difference conditions and Alder’s conjecture. Proc. Natl. Acad. Sci. USA 101 (47), pp. 16417–16418.

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ISSN 1091-6490, FullText, Document, MathReview (Scott D. Ahlgren), ZentralBlatt

Referenced by:

§26.10(iv).

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Bessel Functions. Part IV: Kelvin Functions. Royal Society Mathematical Tables, Volume 10, Cambridge University Press, Cambridge-New York.

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External Links:

MathReview (John Todd), ZentralBlatt

Referenced by:

§10.65(ii), §10.68(iii), §10.68(iv), §10.71(i), 1st item.

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