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1: Bibliography T

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Coulomb functions with complex angular momenta. Comput. Phys. Comm. 17 (4), pp. 351–355.

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Notes:

Single-precision Fortran code.

External Links:

ISSN 0010-4655, Document, Code

Referenced by:

3rd item, 2nd item.

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BibTeX

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Une nouvelle représentation explicite des nombres d’Euler. C. R. Acad. Sci. Paris Sér. A-B 286 (19), pp. A807–A809.

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External Links:

MathReview, ZentralBlatt

Referenced by:

§24.6.

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Rational Chebyshev approximation for the Fermi-Dirac integral $F_{-3/2}(x)$ . Solid–State Electronics 41 (5), pp. 771–773.

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External Links:

Document

Referenced by:

§25.21(vii).

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Further comments on the computation of modified Bessel function ratios. Math. Comp. 35 (151), pp. 937–939.

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External Links:

ISSN 0025-5718, FullText, Document, MathReview (James M. Horner), ZentralBlatt

Referenced by:

§10.74(v).

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Sul comportamento asintotico dell’$n$-esimo polinomio di Laguerre nell’intorno dell’ascissa $4n$ . Comment. Math. Helv. 22, pp. 150–167.

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External Links:

ISSN 0010-2571, Document, MathReview (G. Szegő), ZentralBlatt

Referenced by:

§18.16(iv).

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2: 6 Exponential, Logarithmic, Sine, and
Cosine Integrals

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3: 7 Error Functions, Dawson’s and Fresnel Integrals

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4: Bibliography K

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An adaptive contour code for the numerical evaluation of the oscillatory cuspoid canonical integrals and their derivatives. Computer Physics Comm. 132 (1-2), pp. 142–165.

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External Links:

Document, Code, ZentralBlatt

Referenced by:

§36.15(iii).

Encodings:

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Comment on “The incomplete beta function law for parallel tempering sampling of classical canonical systems” [J. Chem. Phys. 120, 4119 (2004)]. J. Chem. Phys. 121 (2), pp. 1167.

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External Links:

Document

Referenced by:

§8.24(ii).

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A Program for Computing the Conical Functions of the First Kind $P_{-1/2+i\tau }^m(x)$ for $m=0$ and $m=1$ . Comput. Phys. Comm. 23 (1), pp. 51–61.

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Notes:

Single-precision Fortran, maximum accuracy 15D.

External Links:

Document, Code

Referenced by:

§14.31(ii), 1st item.

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Sur le problème de la rotation d’un corps solide autour d’un point fixe. Acta Math. 12 (1), pp. 177–232 (French).

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External Links:

ISSN 0001-5962, Document, MathReview Entry, ZentralBlatt

Referenced by:

§22.19(iv).

Encodings:

BibTeX

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ZEBEC: A mathematical software package for computing simple zeros of Bessel functions of real order and complex argument. Comput. Phys. Comm. 113 (2-3), pp. 220–238.

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Notes:

Double-precision Fortran, maximum accuracy 25S.

External Links:

ISSN 0010-4655, Document, Code, ZentralBlatt

Referenced by:

2nd item.

Encodings:

BibTeX

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5: Bibliography S

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Répartition du courant alternatif dans un conducteur cylindrique de section elliptique. Acad. Roy. Belg. Bull. Cl. Sci. (5) 53 (8), pp. 861–878.

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6th item.

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About von Staudt congruences for Bernoulli numbers. Comment. Math. Univ. St. Paul. 48 (2), pp. 137–144.

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External Links:

ISSN 0010-258X, MathReview (Arnold M. Adelberg), ZentralBlatt

Referenced by:

§24.10(ii).

Encodings:

BibTeX

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Some comments on Fourier analysis, uncertainty and modeling. SIAM Rev. 25 (3), pp. 379–393.

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External Links:

ISSN 0036-1445, Document, MathReview (H. Vincent Poor), ZentralBlatt

Referenced by:

§30.15(ii), §30.15(iii), §30.15(iv), §30.15(v), §30.15(v).

Encodings:

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Effective calculation of the incomplete gamma function for parameter values $\alpha =(2n+1)/2$, $n=0,\mathrm{\dots },5$ . Angew. Informatik 17, pp. 30–32.

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Notes:

Includes Fortran code.

External Links:

ZentralBlatt

Referenced by:

§8.28(ii).

Encodings:

BibTeX

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On the relative extrema of ultraspherical polynomials. Boll. Un. Mat. Ital. (3) 5, pp. 125–127.

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§18.14(iii).

Encodings:

BibTeX

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6: 7.24 Approximations

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Cody (1968) gives minimax rational approximations for the Fresnel integrals (maximum relative precision 19S); for a Fortran algorithm and comments see Snyder (1993).

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Cody et al. (1970) gives minimax rational approximations to Dawson’s integral $F\left(x\right)$ (maximum relative precision 20S–22S).

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Luke (1969b, pp. 323–324) covers $\frac{1}{2}\sqrt{\pi }\mathrm{erf}x$ and ${\mathrm{e}}^{x^2}F\left(x\right)$ for $-3\le x\le 3$ (the Chebyshev coefficients are given to 20D); $\sqrt{\pi }x{\mathrm{e}}^{x^2}\mathrm{erfc}x$ and $2xF\left(x\right)$ for $x\ge 3$ (the Chebyshev coefficients are given to 20D and 15D, respectively). Coefficients for the Fresnel integrals are given on pp. 328–330 (20D).

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Luke (1969b, vol. 2, pp. 422–435) gives main diagonal Padé approximations for $F\left(z\right)$, $\mathrm{erf}z$, $\mathrm{erfc}z$, $C\left(z\right)$, and $S\left(z\right)$; approximate errors are given for a selection of $z$-values.

7: 14.32 Methods of Computation

… ►Essentially the same comments that are made in §15.19 concerning the computation of hypergeometric functions apply to the functions described in the present chapter. In particular, for small or moderate values of the parameters $\mu$ and $\nu$ the power-series expansions of the various hypergeometric function representations given in §§14.3(i)–14.3(iii), 14.19(ii), and 14.20(i) can be selected in such a way that convergence is stable, and reasonably rapid, especially when the argument of the functions is real. In other cases recurrence relations (§14.10) provide a powerful method when applied in a stable direction (§3.6); see Olver and Smith (1983) and Gautschi (1967). … ►

Quadrature (§3.5) of the integral representations given in §§14.12, 14.19(iii), 14.20(iv), and 14.25; see Segura and Gil (1999) and Gil et al. (2000).

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For the computation of conical functions see Gil et al. (2009, 2012), and Dunster (2014).

8: 27 Functions of Number Theory

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9: 7.22 Methods of Computation

… ►The methods available for computing the main functions in this chapter are analogous to those described in §§6.18(i)–6.18(iv) for the exponential integral and sine and cosine integrals, and similar comments apply. Additional references are Matta and Reichel (1971) for the application of the trapezoidal rule, for example, to the first of (7.7.2), and Gautschi (1970) and Cuyt et al. (2008) for continued fractions. … ►For a comprehensive survey of computational methods for the functions treated in this chapter, see van der Laan and Temme (1984, Ch. V).

10: DLMF Project News

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