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21: 7.24 Approximations

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§7.24(i) Approximations in Terms of Elementary Functions

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Cody (1969) provides minimax rational approximations for $\mathrm{erf}x$ and $\mathrm{erfc}x$. The maximum relative precision is about 20S.

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Cody (1968) gives minimax rational approximations for the Fresnel integrals (maximum relative precision 19S); for a Fortran algorithm and comments see Snyder (1993).

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Cody et al. (1970) gives minimax rational approximations to Dawson’s integral $F\left(x\right)$ (maximum relative precision 20S–22S).

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Luke (1969b, vol. 2, pp. 422–435) gives main diagonal Padé approximations for $F\left(z\right)$, $\mathrm{erf}z$, $\mathrm{erfc}z$, $C\left(z\right)$, and $S\left(z\right)$; approximate errors are given for a selection of $z$-values.

22: 25.12 Polylogarithms

… ►The right-hand side is called Clausen’s integral. … ►

Integral Representation

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§25.12(iii) Fermi–Dirac and Bose–Einstein Integrals

►The Fermi–Dirac and Bose–Einstein integrals are defined by … ►In terms of polylogarithms …

23: 22.3 Graphics

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Figure 22.3.13: $\mathrm{sn}(x,k)$ for $k=1-{\mathrm{e}}^{-n}$, $n=0$ to 20, $-5\pi \le x\le 5\pi$. Magnify 3D Help

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Figure 22.3.14: $\mathrm{cn}(x,k)$ for $k=1-{\mathrm{e}}^{-n}$, $n=0$ to 20, $-5\pi \le x\le 5\pi$. Magnify 3D Help

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Figure 22.3.15: $\mathrm{dn}(x,k)$ for $k=1-{\mathrm{e}}^{-n}$, $n=0$ to 20, $-5\pi \le x\le 5\pi$. Magnify 3D Help

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Figure 22.3.28: Density plot of $|\mathrm{sn}(20,k)|$ as a function of complex $k^2$, $-10\le \mathrm{\Re }\left(k^2\right)\le 20$, $-10\le \mathrm{\Im }\left(k^2\right)\le 10$. Grayscale, running from 0 (black) to 10 (white), with $|\mathrm{sn}(20,k)|>10$ truncated to 10. … Magnify

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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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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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Kerimov and Skorokhodov (1984b) tabulates all zeros of the principal values of $K_n\left(z\right)$ and $K_n^{\prime }\left(z\right)$, for $n=2(1)20$, 9S.

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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: 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 …

26: 25.20 Approximations

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Cody et al. (1971) gives rational approximations for $\zeta \left(s\right)$ in the form of quotients of polynomials or quotients of Chebyshev series. The ranges covered are $0.5\le s\le 5$, $5\le s\le 11$, $11\le s\le 25$, $25\le s\le 55$. Precision is varied, with a maximum of 20S.

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Morris (1979) gives rational approximations for ${\mathrm{Li}}_2\left(x\right)$ (§25.12(i)) for $0.5\le x\le 1$. Precision is varied with a maximum of 24S.

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Antia (1993) gives minimax rational approximations for $\mathrm{\Gamma }\left(s+1\right)F_s(x)$, where $F_s(x)$ is the Fermi–Dirac integral (25.12.14), for the intervals and , with $s=-\frac{1}{2},\frac{1}{2},\frac{3}{2},\frac{5}{2}$. For each $s$ there are three sets of approximations, with relative maximum errors ${10}^{-4},{10}^{-8},{10}^{-12}$.

27: 23.14 Integrals

§23.14 Integrals

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23.14.1 $$\int \mathrm{\wp }\left(z\right)dz=-\zeta \left(z\right),$$

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $dx$: differential of $x$ , $\int$: integral, $𝕃$: lattice and $z$: complex

A&S Ref:

18.6.1

Permalink:

http://dlmf.nist.gov/23.14.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §23.14 and Ch.23

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23.14.2 $$\int {\mathrm{\wp }}^2\left(z\right)dz=\frac{1}{6}{\mathrm{\wp }}^{\prime }\left(z\right)+\frac{1}{12}{g}_{2}z,$$

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $g_j$: Weierstrass lattice invariants $g_2$, $g_3$ , $dx$: differential of $x$ , $\int$: integral, $𝕃$: lattice and $z$: complex

A&S Ref:

18.7.1

Permalink:

http://dlmf.nist.gov/23.14.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §23.14 and Ch.23

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23.14.3 $$\int {\mathrm{\wp }}^3\left(z\right)dz=\frac{1}{120}{\mathrm{\wp }}^{\prime \prime \prime }\left(z\right)-\frac{3}{20}{g}_2\zeta \left(z\right)+\frac{1}{10}{g}_{3}z.$$

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Symbols:

$\mathrm{\wp }\left(z\right)$ (= $\mathrm{\wp }\left(z|𝕃\right)$ = $\mathrm{\wp }(z;g_2,g_3)$): Weierstrass $\mathrm{\wp }$-function, $g_j$: Weierstrass lattice invariants $g_2$, $g_3$ , $\zeta \left(z\right)$ (= $\zeta \left(z|𝕃\right)$ = $\zeta (z;g_2,g_3)$): Weierstrass zeta function, $dx$: differential of $x$ , $\int$: integral, $𝕃$: lattice and $z$: complex

A&S Ref:

18.7.2

Permalink:

http://dlmf.nist.gov/23.14.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §23.14 and Ch.23

►For further integrals see Gröbner and Hofreiter (1949, Vol. 1, pp. 161–162), Gradshteyn and Ryzhik (2000, p. 622), and Prudnikov et al. (1990, pp. 51–52).

28: 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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Miller (1946) tabulates $a_k$, ${\mathrm{Ai}}^{\prime }\left(a_k\right)$, $a_k^{\prime }$, $\mathrm{Ai}\left(a_k^{\prime }\right)$, $k=1(1)50$; $b_k$, ${\mathrm{Bi}}^{\prime }\left(b_k\right)$, $b_k^{\prime }$, $\mathrm{Bi}\left(b_k^{\prime }\right)$, $k=1(1)20$. Precision is 8D. Entries for $k=1(1)20$ are reproduced in Abramowitz and Stegun (1964, Chapter 10).

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Sherry (1959) tabulates $a_k$, ${\mathrm{Ai}}^{\prime }\left(a_k\right)$, $a_k^{\prime }$, $\mathrm{Ai}\left(a_k^{\prime }\right)$, $k=1(1)50$; 20S.

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Zhang and Jin (1996, p. 339) tabulates $a_k$, ${\mathrm{Ai}}^{\prime }\left(a_k\right)$, $a_k^{\prime }$, $\mathrm{Ai}\left(a_k^{\prime }\right)$, $b_k$, ${\mathrm{Bi}}^{\prime }\left(b_k\right)$, $b_k^{\prime }$, $\mathrm{Bi}\left(b_k^{\prime }\right)$, $k=1(1)20$; 8D.

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§9.18(v) Integrals

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29: 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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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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Zhang and Jin (1996, p. 642) includes the first 10 zeros of $\mathrm{erf}z$, 9D; the first 25 distinct zeros of $C\left(z\right)$ and $S\left(z\right)$, 8S.

30: Errata

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10.23.11 $$a_k=\frac{1}{2\pi i}{\int }_{|t|=c^{\prime }}f(t)O_k\left(t\right)dt,$$

Originally the contour of integration written incorrectly as $|z|=c^{\prime }$, has been corrected to be $|t|=c^{\prime }$.

Reported by Mark Dunster on 2021-03-22

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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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36.10.14 $$3\left(\frac{{\partial }^2{\mathrm{\Psi }}^{(\mathrm{E})}}{{\partial x}^2}-\frac{{\partial }^2{\mathrm{\Psi }}^{(\mathrm{E})}}{{\partial y}^2}\right)+2\mathrm{i}z\frac{\partial {\mathrm{\Psi }}^{(\mathrm{E})}}{\partial x}-x{\mathrm{\Psi }}^{(\mathrm{E})}=0$$

Originally this equation appeared with $\frac{\partial {\mathrm{\Psi }}^{(\mathrm{H})}}{\partial x}$ in the second term, rather than $\frac{\partial {\mathrm{\Psi }}^{(\mathrm{E})}}{\partial x}$.

Reported 2010-04-02.

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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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The definition of $R_C(x,y)$ was revised in Notations.

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