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11: 20.10 Integrals

§20.10 Integrals

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20.10.1 $${\int }_0^{\mathrm{\infty }}{x}^{s-1}{\theta }_2\left(0\right|\mathrm{i}{x}^2)dx=2^s(1-2^{-s}){\pi }^{-s/2}\mathrm{\Gamma }\left(\frac{1}{2}s\right)\zeta \left(s\right),$$ $\mathrm{\Re }s>1$,

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Symbols:

$\mathrm{\Gamma }\left(z\right)$: gamma function, ${\theta }_j\left(z\right|\tau )$: theta function, $\zeta \left(s\right)$: Riemann zeta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{i}$: imaginary unit, $\int$: integral and $\mathrm{\Re }$: real part

Keywords:

Mellin transform

Referenced by:

§20.10(i), Erratum (V1.1.5) for §20.10(i)

Permalink:

http://dlmf.nist.gov/20.10.E1

Encodings:

pMML, png, TeX

Modification (effective with 1.1.5):

Originally the constraint was $\mathrm{\Re }s>2$. This can be relaxed to $\mathrm{\Re }s>1$ because ${\theta }_2\left(0\right|\mathrm{i}{x}^2)=x^{-1}{\theta }_4\left(0\right|\mathrm{i}{x}^{-2})\sim x^{-1}$ as $x\to 0+$, which follows from (20.7.31) and (20.4.5).

See also:

Annotations for §20.10(i), §20.10 and Ch.20

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20.10.4 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_1\left(\frac{\beta \pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_2\left(\frac{(1+\beta )\pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt=-\frac{\mathrm{\ell }}{\sqrt{s}}\sinh \left(\beta \sqrt{s}\right)\mathrm{sech}\left(\mathrm{\ell }\sqrt{s}\right),$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{sech}z$: hyperbolic secant function, $\sinh z$: hyperbolic sine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $s$: parameter, $\beta$: parameter and $\mathrm{\ell }$: parameter

Permalink:

http://dlmf.nist.gov/20.10.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §20.10(ii), §20.10 and Ch.20

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20.10.5 $${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_3\left(\frac{(1+\beta )\pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt={\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-st}{\theta }_4\left(\frac{\beta \pi }{2\mathrm{\ell }}\right|\frac{\mathrm{i}\pi t}{{\mathrm{\ell }}^2})dt=\frac{\mathrm{\ell }}{\sqrt{s}}\cosh \left(\beta \sqrt{s}\right)\mathrm{csch}\left(\mathrm{\ell }\sqrt{s}\right).$$

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Symbols:

${\theta }_j\left(z\right|\tau )$: theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $dx$: differential of $x$ , $\mathrm{e}$: base of natural logarithm, $\mathrm{csch}z$: hyperbolic cosecant function, $\cosh z$: hyperbolic cosine function, $\mathrm{i}$: imaginary unit, $\int$: integral, $s$: parameter, $\beta$: parameter and $\mathrm{\ell }$: parameter

Permalink:

http://dlmf.nist.gov/20.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §20.10(ii), §20.10 and Ch.20

… ►For further integrals of theta functions see Erdélyi et al. (1954a, pp. 61–62 and 339), Prudnikov et al. (1990, pp. 356–358), Prudnikov et al. (1992a, §3.41), and Gradshteyn and Ryzhik (2000, pp. 627–628).

12: Bibliography G

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Algorithm 471: Exponential integrals. Comm. ACM 16 (12), pp. 761–763.

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Notes:

Includes Algol60 program, maximum accuracy 10S.

External Links:

ISSN 0001-0782, Document

Referenced by:

§6.21(ii), §8.28(vi).

Encodings:

BibTeX

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Exponential integral ${\int }_1^{\mathrm{\infty }}{e}^{-xt}{t}^{-n}𝑑t$ for large values of $n$ . J. Res. Nat. Bur. Standards 62, pp. 123–125.

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External Links:

MathReview (W. Wasow), ZentralBlatt

Referenced by:

§8.11(iii), §8.20(ii).

Encodings:

BibTeX

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A table of integrals of the exponential integral. J. Res. Nat. Bur. Standards Sect. B 73B, pp. 191–210.

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External Links:

MathReview (John Todd), ZentralBlatt

Referenced by:

§6.14(iii), §6.17, §6.7(iv).

Encodings:

BibTeX

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Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:

Includes Fortran program claiming 12S accuracy

External Links:

ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

3rd item, §10.77(ix).

Encodings:

BibTeX

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Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

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External Links:

ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

BibTeX

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13: 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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14: 11.14 Tables

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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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§11.14(iii) Integrals

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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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15: Errata

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27.12.8 $$\frac{\mathrm{li}\left(x\right)}{\varphi \left(m\right)}+O\left(x\exp \left(-\lambda (\alpha ){(\ln x)}^{1/2}\right)\right),$$ $m\le {(\ln x)}^{\alpha }$, $\alpha >0$

Originally the first term was given incorrectly by $\frac{x}{\varphi \left(m\right)}$.

Reported 2017-12-04 by Gergő Nemes.

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9.10.18 $$\mathrm{Ai}\left(z\right)=\frac{3z^{5/4}{\mathrm{e}}^{-(2/3)z^{3/2}}}{4\pi }{\int }_0^{\mathrm{\infty }}\frac{t^{-3/4}{\mathrm{e}}^{-(2/3)t^{3/2}}\mathrm{Ai}\left(t\right)}{z^{3/2}+t^{3/2}}dt$$

The original equation taken from Schulten et al. (1979) was incorrect.

Reported 2015-03-20 by Walter Gautschi.

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9.10.19 $$\mathrm{Bi}\left(x\right)=\frac{3x^{5/4}{\mathrm{e}}^{(2/3)x^{3/2}}}{2\pi }{⨍}_0^{\mathrm{\infty }}\frac{t^{-3/4}{\mathrm{e}}^{-(2/3)t^{3/2}}\mathrm{Ai}\left(t\right)}{x^{3/2}-t^{3/2}}dt$$

The original equation taken from Schulten et al. (1979) was incorrect.

Reported 2015-03-20 by Walter Gautschi.

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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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16: Bibliography F

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Notes:

The “Freely Distributable LIBM” package provides implementations of standard elementary functions plus a few higher functions, e.g. gamma. Double precision, maximum accuracy 20S. Developed by Sun Microsystems.

External Links:

GAMS (package FDLIBM)

Referenced by:

§ ‣ Software Cross Index, § ‣ Software Cross Index.

Encodings:

BibTeX

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Sur certaines sommes des intégral-cosinus. Bull. Soc. Math. Phys. Serbie 12, pp. 13–20 (French).

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External Links:

MathReview (L. Carlitz), ZentralBlatt

Referenced by:

§6.15.

Encodings:

BibTeX

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Tables of Elliptic Integrals of the First, Second, and Third Kind. Technical report Technical Report ARL 64-232, Aerospace Research Laboratories, Wright-Patterson Air Force Base, Ohio.

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Notes:

Reviewed in Math. Comp. v. 1919(1965)509. Table erratum: Math. Comp. v. 20 (1966), no. 96, pp. 639-640.

Referenced by:

§19.37(ii), §19.37(ii), §19.37(iii), §19.37(iii), §19.37(iii).

Encodings:

BibTeX

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A Table of the Complete Elliptic Integral of the First Kind for Complex Values of the Modulus. Part I. Technical report Technical Report ARL 69-0172, Aerospace Research Laboratories, Office of Aerospace Research, Wright-Patterson Air Force Base, Ohio.

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Notes:

Table erratum: Math. Comp. v. 36 (1981), no. 153, p. 318. Part II of this report, with the same date but numbered ARL 69-0173, again puts $k=R\exp (i\theta )$ but with $R$ as parameter and $\theta$ as variable instead of the other way around. Part III, dated May 1970 and numbered ARL 70-0081, contains tables of auxiliary functions to help interpolation.

External Links:

MathReview, ZentralBlatt

Referenced by:

§19.36(ii), §19.37(ii).

Encodings:

BibTeX

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On weighted polynomial approximation on the whole real axis. Acta Math. Acad. Sci. Hungar. 20, pp. 223–225.

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External Links:

ISSN 0001-5954, Document, Link, MathReview (A. K. Varma)

Referenced by:

§18.32.

Encodings:

BibTeX

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17: Bibliography N

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On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.

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External Links:

ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt

Referenced by:

§28.26(ii).

Encodings:

BibTeX

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Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.

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External Links:

FullText, MathReview (I. A. Stegun), ZentralBlatt

Referenced by:

§19.37(iv).

Encodings:

BibTeX

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Error bounds and exponential improvement for Hermite’s asymptotic expansion for the gamma function. Appl. Anal. Discrete Math. 7 (1), pp. 161–179.

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External Links:

ISSN 1452-8630, Document, FullText, MathReview (Junesang Choi), ZentralBlatt

Referenced by:

§5.11(ii), §5.11(ii).

Encodings:

BibTeX

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Error bounds and exponential improvement for the asymptotic expansion of the Barnes $G$-function. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 470 (2172), pp. 20140534, 14.

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External Links:

ISSN 1364-5021, MathReview (Mehdi Hassani), ZentralBlatt

Referenced by:

§5.17, §5.17.

Encodings:

BibTeX

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A table of integrals of the error functions. J. Res. Nat. Bur. Standards Sect B. 73B, pp. 1–20.

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Notes:

See for additions and corrections same journal, Sect. B 75, 149–163 (1975)

External Links:

MathReview, ZentralBlatt

Referenced by:

§7.14(iii), §7.21, §7.7(iii).

Encodings:

BibTeX

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18: Bibliography M

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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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The efficient computation of some generalised exponential integrals. J. Comput. Appl. Math. 148 (2), pp. 363–374.

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External Links:

ISSN 0377-0427, Document, MathReview, ZentralBlatt

Referenced by:

§8.19(v).

Encodings:

BibTeX

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The exponential integral distribution. Statist. Probab. Lett. 5 (3), pp. 209–211.

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External Links:

ISSN 0167-7152, Document, MathReview Entry, ZentralBlatt

Referenced by:

§8.19(xi).

Encodings:

BibTeX

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Tables of Generalized Exponential Integrals. NPL Mathematical Tables, Vol. III, Her Majesty’s Stationery Office, London.

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External Links:

MathReview (L. Fox)

Referenced by:

§8.25(v).

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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19: 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 …

20: 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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