6 Exponential, Logarithmic, Sine, and Cosine Integrals Properties 6.13 Zeros 6.15 Sums

§6.14 Integrals

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Permalink:: http://dlmf.nist.gov/6.14
See also:: Annotations for Ch.6

§6.14(i) Laplace Transforms
§6.14(ii) Other Integrals
§6.14(iii) Compendia

§6.14(i) Laplace Transforms

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Keywords:: Laplace transform, cosine integrals, exponential integrals, integrals, sine integrals
Notes:: See Nielsen (1906b, pp. 48–49: there is a $\frac{1}{2}$ missing in the formula that corresponds to (6.14.2)).
Permalink:: http://dlmf.nist.gov/6.14.i
See also:: Annotations for §6.14 and Ch.6

6.14.1		$\int_{0}^{\infty}e^{-at}E_{1}\left(t\right)\,\mathrm{d}t=\frac{1}{a}\ln\left(1% +a\right),$
$\Re a>-1$ ,
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $E_{1}\left(\NVar{z}\right)$ : exponential integral, $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function and $\Re$ : real part A&S Ref: 5.1.34 (special case of) Permalink: http://dlmf.nist.gov/6.14.E1 Encodings: TeX, pMML, png See also: Annotations for §6.14(i), §6.14 and Ch.6

6.14.2		$\int_{0}^{\infty}e^{-at}\operatorname{Ci}\left(t\right)\,\mathrm{d}t=-\frac{1}% {2a}\ln\left(1+a^{2}\right),$
$\Re a>0$ ,
ⓘ Symbols: $\operatorname{Ci}\left(\NVar{z}\right)$ : cosine integral, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function and $\Re$ : real part A&S Ref: 5.2.28 Referenced by: §6.14(i) Permalink: http://dlmf.nist.gov/6.14.E2 Encodings: TeX, pMML, png See also: Annotations for §6.14(i), §6.14 and Ch.6

6.14.3		$\int_{0}^{\infty}e^{-at}\operatorname{si}\left(t\right)\,\mathrm{d}t=-\frac{1}% {a}\operatorname{arctan}a,$
$\Re a>0$ .
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{arctan}\NVar{z}$ : arctangent function, $\Re$ : real part and $\operatorname{si}\left(\NVar{z}\right)$ : sine integral A&S Ref: 5.2.29 Permalink: http://dlmf.nist.gov/6.14.E3 Encodings: TeX, pMML, png See also: Annotations for §6.14(i), §6.14 and Ch.6

§6.14(ii) Other Integrals

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Keywords:: cosine integrals, exponential integrals, integrals, sine integrals
Notes:: See Nielsen (1906b, pp. 50, 53, and 54: there is a sign error in the formula that corresponds to (6.14.7)).
Permalink:: http://dlmf.nist.gov/6.14.ii
See also:: Annotations for §6.14 and Ch.6

6.14.4		$\int_{0}^{\infty}{E_{1}}^{2}\left(t\right)\,\mathrm{d}t=2\ln 2,$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $E_{1}\left(\NVar{z}\right)$ : exponential integral, $\int$ : integral and $\ln\NVar{z}$ : principal branch of logarithm function A&S Ref: 5.1.33 Permalink: http://dlmf.nist.gov/6.14.E4 Encodings: TeX, pMML, png See also: Annotations for §6.14(ii), §6.14 and Ch.6

6.14.5		$\int_{0}^{\infty}\cos t\operatorname{Ci}\left(t\right)\,\mathrm{d}t=\int_{0}^{% \infty}\sin t\operatorname{si}\left(t\right)\,\mathrm{d}t=-\tfrac{1}{4}\pi,$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\operatorname{Ci}\left(\NVar{z}\right)$ : cosine integral, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\operatorname{si}\left(\NVar{z}\right)$ : sine integral and $\sin\NVar{z}$ : sine function A&S Ref: 5.2.30 Permalink: http://dlmf.nist.gov/6.14.E5 Encodings: TeX, pMML, png See also: Annotations for §6.14(ii), §6.14 and Ch.6

6.14.6		$\int_{0}^{\infty}{\operatorname{Ci}}^{2}\left(t\right)\,\mathrm{d}t=\int_{0}^{% \infty}{\operatorname{si}}^{2}\left(t\right)\,\mathrm{d}t=\tfrac{1}{2}\pi,$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{Ci}\left(\NVar{z}\right)$ : cosine integral, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral and $\operatorname{si}\left(\NVar{z}\right)$ : sine integral A&S Ref: 5.2.31 Permalink: http://dlmf.nist.gov/6.14.E6 Encodings: TeX, pMML, png See also: Annotations for §6.14(ii), §6.14 and Ch.6

6.14.7		$\int_{0}^{\infty}\operatorname{Ci}\left(t\right)\operatorname{si}\left(t\right% )\,\mathrm{d}t=\ln 2.$
ⓘ Symbols: $\operatorname{Ci}\left(\NVar{z}\right)$ : cosine integral, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $\ln\NVar{z}$ : principal branch of logarithm function and $\operatorname{si}\left(\NVar{z}\right)$ : sine integral A&S Ref: 5.2.32 Referenced by: §6.14(ii) Permalink: http://dlmf.nist.gov/6.14.E7 Encodings: TeX, pMML, png See also: Annotations for §6.14(ii), §6.14 and Ch.6

§6.14(iii) Compendia

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Permalink:: http://dlmf.nist.gov/6.14.iii
See also:: Annotations for §6.14 and Ch.6

For collections of integrals, see Apelblat (1983, pp. 110–123), Bierens de Haan (1939, pp. 373–374, 409, 479, 571–572, 637, 664–673, 680–682, 685–697), Erdélyi et al. (1954a, vol. 1, pp. 40–42, 96–98, 177–178, 325), Geller and Ng (1969), Gradshteyn and Ryzhik (2015, §§5.2–5.3 and 6.2–6.27), Marichev (1983, pp. 182–184), Nielsen (1906b), Oberhettinger (1974, pp. 139–141), Oberhettinger (1990, pp. 53–55 and 158–160), Oberhettinger and Badii (1973, pp. 172–179), Prudnikov et al. (1986b, vol. 2, pp. 24–29 and 64–92), Prudnikov et al. (1992a, §§3.4–3.6), Prudnikov et al. (1992b, §§3.4–3.6), and Watrasiewicz (1967).

6.13 Zeros 6.15 Sums

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§6.14 Integrals

Contents

§6.14(i) Laplace Transforms

§6.14(ii) Other Integrals

§6.14(iii) Compendia