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1: 10.75 Tables

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The main tables in Abramowitz and Stegun (1964, Chapter 9) give $J_0\left(x\right)$ to 15D, $J_1\left(x\right)$, $J_2\left(x\right)$, $Y_0\left(x\right)$, $Y_1\left(x\right)$ to 10D, $Y_2\left(x\right)$ to 8D, $x=0(.1)17.5$; $Y_n\left(x\right)-(2/\pi )J_n\left(x\right)\ln x$, $n=0,1$, $x=0(.1)2$, 8D; $J_n\left(x\right)$, $Y_n\left(x\right)$, $n=3(1)9$, $x=0(.2)20$, 5D or 5S; $J_n\left(x\right)$, $Y_n\left(x\right)$, $n=0(1)20(10)50,100$, $x=1,2,5,10,50,100$, 10S; modulus and phase functions $\sqrt{x}{M}_n\left(x\right)$, ${\theta }_n\left(x\right)-x$, $n=0,1,2$, $1/x=0(.01)0.1$, 8D.

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Makinouchi (1966) tabulates all values of $j_{\nu ,m}$ and $y_{\nu ,m}$ in the interval $(0,100)$, with at least 29S. These are for $\nu =0(1)5$, 10, 20; $\nu =\frac{3}{2}$, $\frac{5}{2}$; $\nu =m/n$ with $m=1(1)n-1$ and $n=3(1)8$, except for $\nu =\frac{1}{2}$.

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Bickley et al. (1952) tabulates $x^{-n}{I}_n\left(x\right)$ or ${\mathrm{e}}^{-x}{I}_n\left(x\right)$, $x^n{K}_n\left(x\right)$ or ${\mathrm{e}}^x{K}_n\left(x\right)$, $n=2(1)20$, $x=0$(.01 or .1) 10(.1) 20, 8S; $I_n\left(x\right)$, $K_n\left(x\right)$, $n=0(1)20$, $x=0$ or $0.1(.1)20$, 10S.

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Leung and Ghaderpanah (1979), tabulates all zeros of the principal value of $K_n\left(z\right)$, for $n=2(1)10$, 29S.

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Kerimov and Skorokhodov (1984c) tabulates all zeros of $I_{-n-\frac{1}{2}}\left(z\right)$ and $I_{-n-\frac{1}{2}}^{\prime }\left(z\right)$ in the sector $0\le \mathrm{ph}z\le \frac{1}{2}\pi$ for $n=1(1)20$, 9S.

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2: 25.12 Polylogarithms

… ►When $z={\mathrm{e}}^{\mathrm{i}\theta }$, $0\le \theta \le 2\pi$, (25.12.1) becomes … ►

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Figure 25.12.1: Dilogarithm function ${\mathrm{Li}}_2\left(x\right)$, Magnify

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Figure 25.12.2: Absolute value of the dilogarithm function $\left|{\mathrm{Li}}_2\left(x+\mathrm{i}y\right)\right|$, $-20\le x\le 20$, $-20\le y\le 20$. … Magnify 3D Help

… ►valid when $\mathrm{\Re }s>0$ and , or $\mathrm{\Re }s>1$ and $z=1$. … ►

25.12.12 $${\mathrm{Li}}_s\left(z\right)=\mathrm{\Gamma }\left(1-s\right){\left(\ln \frac{1}{z}\right)}^{s-1}+\sum _{n=0}^{\mathrm{\infty }}\zeta \left(s-n\right)\frac{{(\ln z)}^n}{n!},$$ $s\ne 1,2,3,\mathrm{\dots }$, ,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $!$: factorial (as in $n!$), $\ln z$: principal branch of logarithm function, ${\mathrm{Li}}_s\left(z\right)$: polylogarithm, $n$: nonnegative integer, $s$: complex variable and $z$: complex variable

Keywords:

infinite series, series representation

Source:

Erdélyi et al. (1953a, (1.11.8), p. 29); take $v=1$ and use (25.14.3), (25.11.2)

Permalink:

http://dlmf.nist.gov/25.12.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §25.12(ii), §25.12(ii), §25.12 and Ch.25

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3: Bibliography O

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Hyperasymptotic solutions of higher order linear differential equations with a singularity of rank one. Proc. Roy. Soc. London Ser. A 454, pp. 1–29.

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External Links:

ISSN 1364-5021, Document, MathReview (Donald A. Lutz), ZentralBlatt

Referenced by:

§2.11(v), §2.7(ii).

Encodings:

BibTeX

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An error analysis of the modified Clenshaw method for evaluating Chebyshev and Fourier series. J. Inst. Math. Appl. 20 (3), pp. 379–391.

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External Links:

Document, MathReview (David L. Elliott), ZentralBlatt

Referenced by:

§3.11(ii).

Encodings:

BibTeX

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Error bounds for stationary phase approximations. SIAM J. Math. Anal. 5 (1), pp. 19–29.

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Notes:

See also Olver (1997b)

External Links:

Document, MathReview (L. Berg), ZentralBlatt

Referenced by:

§2.3(iv).

Encodings:

BibTeX

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Modified quotients of cylinder functions. Math. Tables Aids Comput. 10, pp. 27–28.

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External Links:

ISSN 0891-6837, MathReview Entry, ZentralBlatt

Referenced by:

§10.29(i), §10.6(i).

Encodings:

BibTeX

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