Bulirsch%0Aelliptic%20integrals

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21: Bibliography M

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The roots of $J_0(z)-i{J}_1(z)=0$ . Quart. Appl. Math. 47 (2), pp. 375–378.

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External Links:

ISSN 0033-569X, MathReview, ZentralBlatt

Referenced by:

§10.21(ix), 6th item.

Encodings:

BibTeX

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Rational approximations, software and test methods for sine and cosine integrals. Numer. Algorithms 12 (3-4), pp. 259–272.

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Notes:

Includes Fortran code, maximum accuracy 20S.

External Links:

ISSN 1017-1398, Document, MathReview, ZentralBlatt

Referenced by:

§6.13, 4th item, §6.21(ii).

Encodings:

BibTeX

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Über die asymptotische Entwicklung von ${\displaystyle {\int }_0^{-\mathrm{\infty }-i(\arg w-\mu )}}{e}^{\nu z-w\sinh z}𝑑z$, für große Werte von $|w|$ und $|\nu |$. I, II. Proc. Akad. Wet. Amsterdam 35, pp. 1170–1180, 1291–1303 (German).

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External Links:

ISSN 0370-0348, ZentralBlatt

Referenced by:

(11.11.10), (11.11.8), §11.11(iii).

Encodings:

BibTeX

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An improved calculation of the general elliptic integral of the second kind in the neighbourhood of $x=0$ . Numer. Math. 25 (1), pp. 99–101.

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External Links:

ISSN 0029-599X, Document, MathReview (Roland Bulirsch), ZentralBlatt

Referenced by:

§19.36(ii).

Encodings:

BibTeX

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The $q$-analogue of the Laguerre polynomials. J. Math. Anal. Appl. 81 (1), pp. 20–47.

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External Links:

ISSN 0022-247X, Document, MathReview (R. A. Askey), ZentralBlatt

Referenced by:

§18.27(v).

Encodings:

BibTeX

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22: 3.3 Interpolation

… ►where the nodes $x_k=x_0+kh$ ($h>0$) and function $f$ are real, … ►For example, for $k+1$ coincident points the limiting form is given by $\left[z_0,z_0,\mathrm{\dots },z_0\right]f=f^{(k)}(z_0)/k!$. … ►It can be used for solving a nonlinear scalar equation $f(z)=0$ approximately. … ►and with $f=0$ we find that $x=-\mathrm{2.33823\hspace{0.33em}2462}$, with 4 correct digits. … ►For Hermite interpolation, trigonometric interpolation, spline interpolation, rational interpolation (by using continued fractions), interpolation based on Chebyshev points, and bivariate interpolation, see Bulirsch and Rutishauser (1968), Davis (1975, pp. 27–31), and Mason and Handscomb (2003, Chapter 6). …

23: 6.20 Approximations

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Cody and Thacher (1968) provides minimax rational approximations for $E_1\left(x\right)$, with accuracies up to 20S.

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Cody and Thacher (1969) provides minimax rational approximations for $\mathrm{Ei}\left(x\right)$, with accuracies up to 20S.

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MacLeod (1996b) provides rational approximations for the sine and cosine integrals and for the auxiliary functions $\mathrm{f}$ and $\mathrm{g}$, with accuracies up to 20S.

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Luke and Wimp (1963) covers $\mathrm{Ei}\left(x\right)$ for $x\le -4$ (20D), and $\mathrm{Si}\left(x\right)$ and $\mathrm{Ci}\left(x\right)$ for $x\ge 4$ (20D).

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Luke (1969b, pp. 321–322) covers $\mathrm{Ein}\left(x\right)$ and $-\mathrm{Ein}\left(-x\right)$ for $0\le x\le 8$ (the Chebyshev coefficients are given to 20D); $E_1\left(x\right)$ for $x\ge 5$ (20D), and $\mathrm{Ei}\left(x\right)$ for $x\ge 8$ (15D). Coefficients for the sine and cosine integrals are given on pp. 325–327.

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24: 10.75 Tables

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Achenbach (1986) tabulates $J_0\left(x\right)$, $J_1\left(x\right)$, $Y_0\left(x\right)$, $Y_1\left(x\right)$, $x=0(.1)8$, 20D or 18–20S.

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Abramowitz and Stegun (1964, Chapter 11) tabulates ${\int }_0^x{J}_0\left(t\right)dt$, ${\int }_0^x{Y}_0\left(t\right)dt$, $x=0(.1)10$, 10D; ${\int }_0^x{t}^{-1}(1-J_0\left(t\right))dt$, ${\int }_x^{\mathrm{\infty }}{t}^{-1}{Y}_0\left(t\right)dt$, $x=0(.1)5$, 8D.

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Zhang and Jin (1996, p. 270) tabulates ${\int }_0^x{J}_0\left(t\right)dt$, ${\int }_0^x{t}^{-1}(1-J_0\left(t\right))dt$, ${\int }_0^x{Y}_0\left(t\right)dt$, ${\int }_x^{\mathrm{\infty }}{t}^{-1}{Y}_0\left(t\right)dt$, $x=0(.1)1(.5)20$, 8D.

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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, p. 271) tabulates ${\mathrm{e}}^{-x}{\int }_0^x{I}_0\left(t\right)dt$, ${\mathrm{e}}^{-x}{\int }_0^x{t}^{-1}(I_0\left(t\right)-1)dt$, ${\mathrm{e}}^x{\int }_x^{\mathrm{\infty }}{K}_0\left(t\right)dt$, $x{\mathrm{e}}^x{\int }_x^{\mathrm{\infty }}{t}^{-1}{K}_0\left(t\right)dt$, $x=0(.1)1(.5)20$, 8D.

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25: 11.14 Tables

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Abramowitz and Stegun (1964, Chapter 12) tabulates ${𝐇}_n\left(x\right)$, ${𝐇}_n\left(x\right)-Y_n\left(x\right)$, and $I_n\left(x\right)-{𝐋}_n\left(x\right)$ for $n=0,1$ and $x=0(.1)5$, $x^{-1}=0(.01)0.2$ to 6D or 7D.

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Agrest et al. (1982) tabulates ${𝐇}_n\left(x\right)$ and ${\mathrm{e}}^{-x}{𝐋}_n\left(x\right)$ for $n=0,1$ and $x=0(.001)5(.005)15(.01)100$ to 11D.

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Zhang and Jin (1996) tabulates ${𝐇}_n\left(x\right)$ and ${𝐋}_n\left(x\right)$ for $n=-4(1)3$ and $x=0(1)20$ to 8D or 7S.

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Abramowitz and Stegun (1964, Chapter 12) tabulates ${\int }_0^x(I_0\left(t\right)-{𝐋}_0\left(t\right))dt$ and $(2/\pi ){\int }_x^{\mathrm{\infty }}{t}^{-1}{𝐇}_0\left(t\right)dt$ for $x=0(.1)5$ to 5D or 7D; ${\int }_0^x({𝐇}_0\left(t\right)-Y_0\left(t\right))dt-(2/\pi )\ln x$, ${\int }_0^x(I_0\left(t\right)-{𝐋}_0\left(t\right))dt-(2/\pi )\ln x$, and ${\int }_x^{\mathrm{\infty }}{t}^{-1}({𝐇}_0\left(t\right)-Y_0\left(t\right))dt$ for $x^{-1}=0(.01)0.2$ to 6D.

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Agrest et al. (1982) tabulates ${\int }_0^x{𝐇}_0\left(t\right)dt$ and ${\mathrm{e}}^{-x}{\int }_0^x{𝐋}_0\left(t\right)dt$ for $x=0(.001)5(.005)15(.01)100$ to 11D.

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26: 33.24 Tables

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Abramowitz and Stegun (1964, Chapter 14) tabulates $F_0(\eta ,\rho )$, $G_0(\eta ,\rho )$, $F_0^{\prime }(\eta ,\rho )$, and $G_0^{\prime }(\eta ,\rho )$ for $\eta =0.5(.5)20$ and $\rho =1(1)20$, 5S; $C_0\left(\eta \right)$ for $\eta =0(.05)3$, 6S.

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Curtis (1964a) tabulates $P_{\mathrm{\ell }}(ϵ,r)$, $Q_{\mathrm{\ell }}(ϵ,r)$ (§33.1), and related functions for $\mathrm{\ell }=0,1,2$ and $ϵ=-2(.2)2$, with $x=0(.1)4$ for and $x=0(.1)10$ for $ϵ\ge 0$; 6D.

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27: 7.23 Tables

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Abramowitz and Stegun (1964, Chapter 7) includes $\mathrm{erf}x$, $(2/\sqrt{\pi }){\mathrm{e}}^{-x^2}$, $x\in [0,2]$, 10D; $(2/\sqrt{\pi }){\mathrm{e}}^{-x^2}$, $x\in [2,10]$, 8S; $x{\mathrm{e}}^{x^2}\mathrm{erfc}x$, $x^{-2}\in [0,0.25]$, 7D; $2^n\mathrm{\Gamma }\left(\frac{1}{2}n+1\right){\mathrm{i}}^n\mathrm{erfc}\left(x\right)$, $n=1(1)6,10,11$, $x\in [0,5]$, 6S; $F\left(x\right)$, $x\in [0,2]$, 10D; $xF\left(x\right)$, $x^{-2}\in [0,0.25]$, 9D; $C\left(x\right)$, $S\left(x\right)$, $x\in [0,5]$, 7D; $\mathrm{f}\left(x\right)$, $\mathrm{g}\left(x\right)$, $x\in [0,1]$, $x^{-1}\in [0,1]$, 15D.

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Abramowitz and Stegun (1964, Table 27.6) includes the Goodwin–Staton integral $G\left(x\right)$, $x=1(.1)3(.5)8$, 4D; also $G\left(x\right)+\ln x$, $x=0(.05)1$, 4D.

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Zhang and Jin (1996, pp. 637, 639) includes $(2/\sqrt{\pi }){\mathrm{e}}^{-x^2}$, $\mathrm{erf}x$, $x=0(.02)1(.04)3$, 8D; $C\left(x\right)$, $S\left(x\right)$, $x=0(.2)10(2)100(100)500$, 8D.

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Abramowitz and Stegun (1964, Chapter 7) includes $w\left(z\right)$, $x=0(.1)3.9$, $y=0(.1)3$, 6D.

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Zhang and Jin (1996, pp. 638, 640–641) includes the real and imaginary parts of $\mathrm{erf}z$, $x\in [0,5]$, $y=0.5(.5)3$, 7D and 8D, respectively; the real and imaginary parts of ${\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{\pm \mathrm{i}{t}^2}dt$, $(1/\sqrt{\pi }){\mathrm{e}}^{\mp \mathrm{i}(x^2+(\pi /4))}{\int }_x^{\mathrm{\infty }}{\mathrm{e}}^{\pm \mathrm{i}{t}^2}dt$, $x=0(.5)20(1)25$, 8D, together with the corresponding modulus and phase to 8D and 6D (degrees), respectively.

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28: 6.16 Mathematical Applications

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§6.16(i) The Gibbs Phenomenon

… ►Hence, if $x$ is fixed and $n\to \mathrm{\infty }$, then $S_n(x)\to \frac{1}{4}\pi$, $0$, or $-\frac{1}{4}\pi$ according as , $x=0$, or ; compare (6.2.14). … ►The first maximum of $\frac{1}{2}\mathrm{Si}\left(x\right)$ for positive $x$ occurs at $x=\pi$ and equals $(1.1789\mathrm{\dots })\times \frac{1}{4}\pi$; compare Figure 6.3.2. … ►

§6.16(ii) Number-Theoretic Significance of $\mathrm{li}\left(x\right)$

►If we assume Riemann’s hypothesis that all nonreal zeros of $\zeta \left(s\right)$ have real part of $\frac{1}{2}$ (§25.10(i)), then …

29: 9.18 Tables

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Zhang and Jin (1996, p. 337) tabulates $\mathrm{Ai}\left(x\right)$, ${\mathrm{Ai}}^{\prime }\left(x\right)$, $\mathrm{Bi}\left(x\right)$, ${\mathrm{Bi}}^{\prime }\left(x\right)$ for $x=0(1)20$ to 8S and for $x=-20(1)0$ to 9D.

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Rothman (1954b) tabulates ${\int }_0^x\mathrm{Ai}\left(t\right)dt$ and ${\int }_0^x\mathrm{Bi}\left(t\right)dt$ for $x=-10(.1)\mathrm{\infty }$ and $-10(.1)2$, respectively; 7D. The entries in the columns headed ${\int }_0^x\mathrm{Ai}\left(-x\right)dx$ and ${\int }_0^x\mathrm{Bi}\left(-x\right)dx$ all have the wrong sign. The tables are reproduced in Abramowitz and Stegun (1964, Chapter 10), and the sign errors are corrected in later reprintings.

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National Bureau of Standards (1958) tabulates ${\int }_0^x\mathrm{Ai}\left(-t\right)dt$ and ${\int }_0^x{\int }_0^v\mathrm{Ai}\left(-t\right)dtdv$ (see (9.10.20)) for $x=-2(.01)5$ to 8D and 7D, respectively.

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Scorer (1950) tabulates $\mathrm{Gi}\left(x\right)$ and $\mathrm{Hi}\left(-x\right)$ for $x=0(.1)10$; 7D.

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Rothman (1954a) tabulates ${\int }_0^x\mathrm{Gi}\left(t\right)dt$, ${\mathrm{Gi}}^{\prime }\left(x\right)$, ${\int }_0^x\mathrm{Hi}\left(-t\right)dt$, $-{\mathrm{Hi}}^{\prime }\left(-x\right)$ for $x=0(.1)10$; 7D.

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30: 25.12 Polylogarithms

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

Integral Representation

… ►valid when $\mathrm{\Re }s>0$, $\mathrm{\Im }a>0$ or $\mathrm{\Re }s>1$, $\mathrm{\Im }a=0$. … ►The Fermi–Dirac and Bose–Einstein integrals are defined by … ►In terms of polylogarithms …