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1: 15.4 Special Cases

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§15.4(iii) Other Arguments

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2: Bibliography T

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COULCC: A continued-fraction algorithm for Coulomb functions of complex order with complex arguments. Comput. Phys. Comm. 36 (4), pp. 363–372.

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Notes:

Double-precision Fortran, minimum accuracy: 14D. See also Thompson (2004)

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Document, Code, ZentralBlatt

Referenced by:

1st item, §33.23(vi), 3rd item.

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Erratum to “COULCC: A continued-fraction algorithm for Coulomb functions of complex order with complex arguments”. Comput. Phys. Comm. 159 (3), pp. 241–242.

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Referenced by:

§33.23(vi), I. J. Thompson and A. R. Barnett (1985).

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3: Bibliography C

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Tables of Jacobian Elliptic Functions Whose Arguments are Rational Fractions of the Quarter Period. National Physical Laboratory Mathematical Tables, Vol. 7, Her Majesty’s Stationery Office, London.

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ISBN 00-791-1158-0, MathReview (John Todd), ZentralBlatt

Referenced by:

§22.20(vii), §22.21.

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4: 35.1 Special Notation

… ►All fractional or complex powers are principal values. ► ►►

$a,b$	complex variables.
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►The main functions treated in this chapter are the multivariate gamma and beta functions, respectively ${\mathrm{\Gamma }}_m\left(a\right)$ and ${\mathrm{B}}_m(a,b)$, and the special functions of matrix argument: Bessel (of the first kind) $A_{\nu }\left(𝐓\right)$ and (of the second kind) $B_{\nu }\left(𝐓\right)$; confluent hypergeometric (of the first kind) ${}_1{}^{}F_1^{}(a;b;𝐓)$ or ${}_1{}^{}F_1^{}({\displaystyle \genfrac{}{}{0pt}{}{a}{b}};𝐓)$ and (of the second kind) $\mathrm{\Psi }(a;b;𝐓)$; Gaussian hypergeometric ${}_2{}^{}F_1^{}(a_1,a_2;b;𝐓)$ or ${}_2{}^{}F_1^{}({\displaystyle \genfrac{}{}{0pt}{}{a_1,a_2}{b}};𝐓)$; generalized hypergeometric ${}_p{}^{}F_q^{}(a_1,\mathrm{\dots },a_p;b_1,\mathrm{\dots },b_q;𝐓)$ or ${}_p{}^{}F_q^{}({\displaystyle \genfrac{}{}{0pt}{}{a_1,\mathrm{\dots },a_p}{b_1,\mathrm{\dots },b_q}};𝐓)$. ►An alternative notation for the multivariate gamma function is ${\mathrm{\Pi }}_m(a)={\mathrm{\Gamma }}_m\left(a+\frac{1}{2}(m+1)\right)$ (Herz (1955, p. 480)). Related notations for the Bessel functions are ${𝒥}_{\nu +\frac{1}{2}(m+1)}(𝐓)=A_{\nu }\left(𝐓\right)/A_{\nu }\left(𝟎\right)$ (Faraut and Korányi (1994, pp. 320–329)), $K_m(0,\mathrm{\dots },0,\nu |𝐒,𝐓)={\left|𝐓\right|}^{\nu }{B}_{\nu }\left(𝐒𝐓\right)$ (Terras (1988, pp. 49–64)), and ${𝒦}_{\nu }(𝐓)={\left|𝐓\right|}^{\nu }{B}_{\nu }\left(𝐒𝐓\right)$ (Faraut and Korányi (1994, pp. 357–358)).

5: Bibliography S

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A code to evaluate modified Bessel functions based on the continued fraction method. Comput. Phys. Comm. 105 (2-3), pp. 263–272.

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

Referenced by:

7th item.

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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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Inequalities involving cylindrical functions of nearly equal argument and order. Proc. Amer. Math. Soc. 5 (3), pp. 337–344.

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ISSN 0002-9939, FullText, MathReview (N. D. Kazarinoff), ZentralBlatt

Referenced by:

§10.14.

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An inequality involving Bessel functions of argument nearly equal to their order. Proc. Amer. Math. Soc. 4 (6), pp. 858–859.

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ISSN 0002-9939, FullText, MathReview (N. D. Kazarinoff), ZentralBlatt

Referenced by:

§10.14.

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Root-rational-fraction package for exact calculation of vector-coupling coefficients. Comput. Phys. Comm. 21 (2), pp. 195–205.

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Notes:

Double-precision Fortran, maximum accuracy 16D

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Document, Code

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10th item.

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6: 14.32 Methods of Computation

… ►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). … ►

Evaluation (§3.10) of the continued fractions given in §14.14. See Gil and Segura (2000).

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7: Bibliography M

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Two-point quasi-fractional approximations to the Airy function $\mathrm{Ai}(x)$ . J. Comput. Phys. 99 (2), pp. 337–340.

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Notes:

There is an error in Eq. (5): the second term in parentheses needs to be multiplied by $x$.

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ISSN 0021-9991, Document, MathReview Entry, ZentralBlatt

Referenced by:

1st item.

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An Introduction to the Fractional Calculus and Fractional Differential Equations. A Wiley-Interscience Publication, John Wiley & Sons, Inc., New York.

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ISBN 0-471-58884-9, MathReview (K. C. Gupta), ZentralBlatt

Referenced by:

§1.15(vi), Erratum (V1.0.14) for Subsections 1.15(vi), 1.15(vii), 2.6(iii).

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The dilogarithm function of a real argument. Math. Comp. 33 (146), pp. 778–787.

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ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

2nd item, 4th item.

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A new Stirling series as continued fraction. Numer. Algorithms 56 (1), pp. 17–26.

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ISSN 1017-1398, Document, FullText, MathReview, ZentralBlatt

Referenced by:

§5.10.

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A continued fraction approximation of the gamma function. J. Math. Anal. Appl. 402 (2), pp. 405–410.

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ISSN 0022-247X, Document, FullText, MathReview (Paul Levrie), ZentralBlatt

Referenced by:

§5.10.

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8: 33.8 Continued Fractions

§33.8 Continued Fractions

►With arguments $\eta ,\rho$ suppressed, … ►

$a=1+\mathrm{\ell }\pm \mathrm{i}\eta ,$

… ►The continued fraction (33.8.1) converges for all finite values of $\rho$, and (33.8.2) converges for all $\rho \ne 0$. … ►The ambiguous sign in (33.8.4) has to agree with that of the final denominator in (33.8.1) when the continued fraction has converged to the required precision. …

9: 16.4 Argument Unity

§16.4 Argument Unity

… ►The function ${}_{q+1}{}^{}F_q^{}$ with argument unity and general values of the parameters is discussed in Bühring (1992). … ►with limiting form $a\left(\psi \left(a+n+1\right)-\psi \left(a\right)\right)=\frac{a\frac{d}{da}{\left(a\right)}_{n+1}}{{\left(a\right)}_{n+1}}$ in the case that $c=a+1$. … ►

§16.4(iv) Continued Fractions

►For continued fractions for ratios of ${}_3{}^{}F_2^{}$ functions with argument unity, see Cuyt et al. (2008, pp. 315–317). …

10: 7.22 Methods of Computation

… ►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. … ►The computation of these functions can be based on algorithms for the complementary error function with complex argument; compare (7.19.3). … ►For a comprehensive survey of computational methods for the functions treated in this chapter, see van der Laan and Temme (1984, Ch. V).