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1: Bibliography B

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Algorithms for computing Bessel functions of half-integer order with complex arguments. Zh. Vychisl. Mat. i Mat. Fiz. 28 (10), pp. 1449–1460, 1597.

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Notes:

English translation in U.S.S.R. Comput. Math. and Math. Phys. 28(1988), no. 5, 109–117

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ISSN 0044-4669, Document, MathReview (Walter Gautschi), ZentralBlatt

Referenced by:

§10.77(iv).

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Avoided crossings of the quartic oscillator. J. Phys. A 30 (9), pp. 3057–3067.

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ISSN 0305-4470, Document, MathReview (Gabriel Alvarez), ZentralBlatt

Referenced by:

§31.17(ii).

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Prolate angular spheroidal wave functions. Comput. Phys. Comm. 30 (2), pp. 187–192.

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Document, Code

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1st item.

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Prolate radial spheroidal wave functions. Comput. Phys. Comm. 30 (2), pp. 177–185.

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Document, Code

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1st item.

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More than 41% of the zeros of the zeta function are on the critical line. Acta Arith. 150 (1), pp. 35–64.

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ISSN 0065-1036, Document, Link, MathReview (Timothy S. Trudgian), ZentralBlatt

Referenced by:

§25.10(ii), §25.10(ii), Erratum (V1.1.4) for Subsection 25.10(ii).

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2: 35.12 Software

… ►For an algorithm to evaluate zonal polynomials, and an implementation of the algorithm in Maple by Zeilberger, see Lapointe and Vinet (1996).

3: 25.10 Zeros

… ►More than 41% of all the zeros in the critical strip lie on the critical line (Bui et al. (2011)). …

4: Bibliography L

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Exact operator solution of the Calogero-Sutherland model. Comm. Math. Phys. 178 (2), pp. 425–452.

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Notes:

For a Maple implementation of the algorithm given in this paper for Jack and zonal polynomials see Zeilberger (website).

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ISSN 0010-3616, Link, MathReview (Kazuhiro Hikami), ZentralBlatt

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§35.12.

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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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Algorithm 344: Student’s t-distribution [S14]. Comm. ACM 12 (1), pp. 37–38.

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Notes:

Single-precision Fortran, maximum accuracy 8S. For Remarks see same journal 13(1970), pp. 124 and 449.

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ISSN 0001-0782, Document, Remark 1 and Remark 2

Referenced by:

§8.28(iv).

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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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Numerical Evaluation of Special Functions. In Mathematics of Computation 1943–1993: A Half-Century of Computational Mathematics (Vancouver, BC, 1993), Proc. Sympos. Appl. Math., Vol. 48, pp. 79–125.

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Notes:

An updated version exists online.

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FullText, MathReview (John P. Coleman), ZentralBlatt

Referenced by:

Ch.3.

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5: Bibliography Z

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Remark on “Algorithm 916: computing the Faddeyeva and Voigt functions”: efficiency improvements and Fortran translation. ACM Trans. Math. Softw. 42 (3), pp. 26:1–26:9.

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Notes:

Includes MATLAB and Fortran programs claiming 6S or 13S accuracy for single and double precision

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ISSN 0098-3500, Link, Document, MathReview Entry

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3rd item, 6th item, §7.25(iii), §7.25(vi).

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Doron Zeilberger’s Maple Packages and Programs Department of Mathematics, Rutgers University, New Jersey.

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Notes:

Includes hypergeometric and $q$-hypergeometric summation, solution of difference equations (for example, by Petkovšek’s algorithm), combinatorial algorithms, and (package LUC) evaluation of zonal polynomials.

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Home Page

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index, L. Lapointe and L. Vinet (1996).

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Tables of the Legendre functions $P_{-1/2+i\tau }(x)$. Part I. Translated by D. E. Brown. Mathematical Tables Series, Vol. 22, Pergamon Press, Oxford.

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Notes:

Translation of Žurina and Karmazina, Tablitsy funktsii Lezhandra $P_{-1/2+i\tau }(x)$. Tom I, Izdat. Akad. Nauk SSSR, Moscow, 1960.

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MathReview (John Todd), ZentralBlatt

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4th item.

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Tables of the Legendre functions $P_{-1/2+i\tau }(x)$. Part II. Translated by Prasenjit Basu. Mathematical Tables Series, Vol. 38. A Pergamon Press Book, The Macmillan Co., New York.

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Notes:

Translation of Žurina and Karmazina, Tablitsy funktsii Lezhandra $P_{-1/2+i\tau }(x)$. Tom II, Izdat. Akad. Nauk SSSR, Moscow, 1962.

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MathReview, ZentralBlatt

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4th item.

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Tablitsy vyrozhdennoi gipergeometricheskoi funktsii. Vyčisl. Centr Akad. Nauk SSSR, Moscow (Russian).

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Notes:

Translated title: Tables of the confluent hypergeometric function

External Links:

MathReview (A. H. Stroud), ZentralBlatt

Referenced by:

1st item.

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6: 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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External Links:

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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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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Fast computation of incomplete elliptic integral of first kind by half argument transformation. Numer. Math. 116 (4), pp. 687–719.

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ISSN 0029-599X, Document, FullText, MathReview (Mehdi Hassani), ZentralBlatt

Referenced by:

§19.36(iv), §19.36(iv).

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7: 22.4 Periods, Poles, and Zeros

… ►Using the p,q notation of (22.2.10), Figure 22.4.2 serves as a mnemonic for the poles, zeros, periods, and half-periods of the 12 Jacobian elliptic functions as follows. …(b) The difference between p and the nearest q is a half-period of $\mathrm{p}\mathrm{q}(z,k)$. This half-period will be plus or minus a member of the triple $K,\mathrm{i}{K}^{\prime },K+\mathrm{i}{K}^{\prime }$; the other two members of this triple are quarter periods of $\mathrm{p}\mathrm{q}(z,k)$. ►

§22.4(iii) Translation by Half or Quarter Periods

►See Table 22.4.3. …

8: Sidebar 5.SB1: Gamma & Digamma Phase Plots

… ►The color encoded phases of $\mathrm{\Gamma }\left(z\right)$ (above) and $\psi \left(z\right)$ (below), are constrasted in the negative half of the complex plane. ►In the upper half of the image, the poles of $\mathrm{\Gamma }\left(z\right)$ are clearly visible at negative integer values of $z$: the phase changes by $2\pi$ around each pole, showing a full revolution of the color wheel. … ►In the lower half of the image, the poles of $\psi \left(z\right)$ (corresponding to the poles of $\mathrm{\Gamma }\left(z\right)$) and the zeros between them are clear. …

9: 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

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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

External Links:

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

Referenced by:

§27.12.

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Parabolic cylinder functions of integer and half-integer orders for nonnegative arguments. Comput. Phys. Comm. 115 (1), pp. 69–86.

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Notes:

Double-precision Fortran, minimum accuracy 14S, maximum accuracy 18D.

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ISSN 0010-4655, Document, Code, MathReview, ZentralBlatt

Referenced by:

3rd item.

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

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Notes:

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

External Links:

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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10: Errata

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Version 1.1.6 (June 30, 2022)

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In the paragraph immediately below (25.10.4), it was originally stated that “more than one-third of all zeros in the critical strip lie on the critical line.” which referred to Levinson (1974). This sentence has been updated with “one-third” being replaced with “41%” now referring to Bui et al. (2011) (suggested by Gergő Nemes on 2021-08-23).

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Clarifications were made to this subsection with the addition of Equations (3.5.30_5), (3.5.33_1), (3.5.33_2), (3.5.33_3) and Table 3.5.17_5.

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There has been disagreement about the identification of the Chebyshev polynomials of the third and fourth kinds, denoted $V_n\left(x\right)$ and $W_n\left(x\right)$, in published references. Originally, DLMF used the definitions given in (Andrews et al., 1999, Remark 2.5.3). However, those definitions were the reverse of those used by Mason and Handscomb (2003), Gautschi (2004) following Mason (1993) and Gautschi (1992), as was noted in several warnings added in Version 1.0.10 (August 7, 2015) of the DLMF. Since the latter definitions are more widely established, the DLMF is now adopting the definitions of Mason and Handscomb (2003). Essentially, what we previously denoted $V_n(x)$ is now written as $W_n(x)$, and vice-versa.

This notational interchange necessitated changes in Tables 18.3.1, 18.5.1, and 18.6.1, and in Equations (18.5.3), (18.5.4), (18.7.5), (18.7.6), (18.7.17), (18.7.18), (18.9.11), and (18.9.12).

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Special cases of normalization of Jacobi polynomials for which the general formula is undefined have been stated explicitly in Table 18.3.1.

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