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11: 10.75 Tables

… ►Also, for additional listings of tables pertaining to complex arguments see Babushkina et al. (1997). … ►

Zhang and Jin (1996, pp. 185–195) tabulates $J_n\left(x\right)$, $J_n^{\prime }\left(x\right)$, $Y_n\left(x\right)$, $Y_n^{\prime }\left(x\right)$, $n=0(1)10(10)50,100$, $x=1$, 5, 10, 25, 50, 100, 9S; $J_{n+\alpha }\left(x\right)$, $J_{n+\alpha }^{\prime }\left(x\right)$, $Y_{n+\alpha }\left(x\right)$, $Y_{n+\alpha }^{\prime }\left(x\right)$, $n=0(1)5,10,30,50,100$, $\alpha =\frac{1}{4},\frac{1}{3},\frac{1}{2},\frac{2}{3},\frac{3}{4}$, $x=1,5,10,50$, 8S; real and imaginary parts of $J_{n+\alpha }\left(z\right)$, $J_{n+\alpha }^{\prime }\left(z\right)$, $Y_{n+\alpha }\left(z\right)$, $Y_{n+\alpha }^{\prime }\left(z\right)$, $n=0(1)15,20(10)50,100$, $\alpha =0,\frac{1}{2}$, $z=4+2\mathrm{i}$, $20+10\mathrm{i}$, 8S.

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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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Zhang and Jin (1996, pp. 240–250) tabulates $I_n\left(x\right)$, $I_n^{\prime }\left(x\right)$, $K_n\left(x\right)$, $K_n^{\prime }\left(x\right)$, $n=0(1)10(10)50,100$, $x=1,5,10,25,50,100$, 9S; $I_{n+\alpha }\left(x\right)$, $I_{n+\alpha }^{\prime }\left(x\right)$, $K_{n+\alpha }\left(x\right)$, $K_{n+\alpha }^{\prime }\left(x\right)$, $n=0(1)5$, 10, 30, 50, 100, $\alpha =\frac{1}{4}$, $\frac{1}{3}$, $\frac{1}{2}$, $\frac{2}{3}$, $\frac{3}{4}$, $x=1$, 5, 10, 50, 8S; real and imaginary parts of $I_{n+\alpha }\left(z\right)$, $I_{n+\alpha }^{\prime }\left(z\right)$, $K_{n+\alpha }\left(z\right)$, $K_{n+\alpha }^{\prime }\left(z\right)$, $n=0(1)15$, 20(10)50, 100, $\alpha =0,\frac{1}{2}$, $z=4+2\mathrm{i},20+10\mathrm{i}$, 8S.

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Zhang and Jin (1996, pp. 296–305) tabulates ${𝗃}_n\left(x\right)$, ${𝗃}_n^{\prime }\left(x\right)$, ${𝗒}_n\left(x\right)$, ${𝗒}_n^{\prime }\left(x\right)$, ${𝗂}_n^{(1)}\left(x\right)$, $𝗂_n^{(1)}{}_{}{}^{\prime }\left(x\right)$, ${𝗄}_n\left(x\right)$, ${𝗄}_n^{\prime }\left(x\right)$, $n=0(1)10(10)30$, 50, 100, $x=1$, 5, 10, 25, 50, 100, 8S; $x{𝗃}_n\left(x\right)$, ${(x{𝗃}_n\left(x\right))}^{\prime }$, $x{𝗒}_n\left(x\right)$, ${(x{𝗒}_n\left(x\right))}^{\prime }$ (Riccati–Bessel functions and their derivatives), $n=0(1)10(10)30$, 50, 100, $x=1$, 5, 10, 25, 50, 100, 8S; real and imaginary parts of ${𝗃}_n\left(z\right)$, ${𝗃}_n^{\prime }\left(z\right)$, ${𝗒}_n\left(z\right)$, ${𝗒}_n^{\prime }\left(z\right)$, ${𝗂}_n^{(1)}\left(z\right)$, $𝗂_n^{(1)}{}_{}{}^{\prime }\left(z\right)$, ${𝗄}_n\left(z\right)$, ${𝗄}_n^{\prime }\left(z\right)$, $n=0(1)15$, 20(10)50, 100, $z=4+2\mathrm{i}$, $20+10\mathrm{i}$, 8S. (For the notation replace $j,y,\mathrm{i},k$ by $𝗃$, $𝗒$, ${𝗂}^{(1)}$, $𝗄$, respectively.)

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12: Errata

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Originally the matrix in the argument of the Gaussian hypergeometric function of matrix argument ${}_2{}^{}F_1^{}$ was written with round brackets. This matrix has been rewritten with square brackets to be consistent with the rest of the DLMF.

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The arguments of some of the functions in (33.11.2)–(33.11.7) were included to improve clarity of the presentation.

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10.17.14 $$\left|R_{\mathrm{\ell }}^{\pm }(\nu ,z)\right|\le 2|a_{\mathrm{\ell }}(\nu )|{𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)\exp \left(|{\nu }^2-\frac{1}{4}|{𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-1}\right)\right)$$

Originally the factor ${𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-1}\right)$ in the argument to the exponential was written incorrectly as ${𝒱}_{z,\pm \mathrm{i}\mathrm{\infty }}\left(t^{-\mathrm{\ell }}\right)$.

Reported 2014-09-27 by Gergő Nemes.

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Several new equations have been added. See (8.17.24), (20.7.34), §20.11(v), (26.12.27), (36.2.28), and (36.2.29).

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Bibliographic citations were added in §§1.13(v), 10.14, 10.21(ii), 18.15(v), 18.32, 30.16(iii), 32.13(ii), and as general references in Chapters 19, 20, 22, and 23.

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