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21: 8.25 Methods of Computation

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§8.25(iv) Continued Fractions

►The computation of $\gamma (a,z)$ and $\mathrm{\Gamma }(a,z)$ by means of continued fractions is described in Jones and Thron (1985) and Gautschi (1979b, §§4.3, 5). …

22: 33.8 Continued Fractions

§33.8 Continued Fractions

… ►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. …

23: 14.14 Continued Fractions

§14.14 Continued Fractions

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24: 31.18 Methods of Computation

… ►The computation of the accessory parameter for the Heun functions is carried out via the continued-fraction equations (31.4.2) and (31.11.13) in the same way as for the Mathieu, Lamé, and spheroidal wave functions in Chapters 28–30.

25: Annie A. M. Cuyt

… ►Subsequently she was a Research fellow with the Alexander von Humboldt Foundation (Germany), she obtained the Habilitation (1986) and became author or co-author of several books, including Handbook of Continued Fractions for Special Functions. …

26: 14.32 Methods of Computation

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

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27: 15.7 Continued Fractions

§15.7 Continued Fractions

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28: 33.23 Methods of Computation

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§33.23(v) Continued Fractions

►§33.8 supplies continued fractions for $F_{\mathrm{\ell }}^{\prime }/F_{\mathrm{\ell }}$ and $H_{\mathrm{\ell }}^{\pm }{}_{}{}^{\prime }/H_{\mathrm{\ell }}^{\pm }$. … ►Thompson and Barnett (1985, 1986) and Thompson (2004) use combinations of series, continued fractions, and Padé-accelerated asymptotic expansions (§3.11(iv)) for the analytic continuations of Coulomb functions. …

29: 28.15 Expansions for Small $q$

… ►Higher coefficients can be found by equating powers of $q$ in the following continued-fraction equation, with $a={\lambda }_{\nu }\left(q\right)$: ►

28.15.2 $$a-{\nu }^2-\frac{q^2}{a-{(\nu +2)}^2-}\frac{q^2}{a-{(\nu +4)}^2-}\mathrm{\cdots }=\frac{q^2}{a-{(\nu -2)}^2-}\frac{q^2}{a-{(\nu -4)}^2-}\mathrm{\cdots }.$$

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Symbols:

$q=h^2$: parameter, $\nu$: complex parameter and $a$: parameter

Referenced by:

item (e)

Permalink:

http://dlmf.nist.gov/28.15.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §28.15(i), §28.15 and Ch.28

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30: 15.19 Methods of Computation

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§15.19(v) Continued Fractions

►In Colman et al. (2011) an algorithm is described that uses expansions in continued fractions for high-precision computation of the Gauss hypergeometric function, when the variable and parameters are real and one of the numerator parameters is a positive integer. …