6 Exponential, Logarithmic, Sine, and Cosine Integrals Properties 6.6 Power Series 6.8 Inequalities

§6.7 Integral Representations

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Keywords:: exponential integrals, integral representations
Permalink:: http://dlmf.nist.gov/6.7
See also:: Annotations for Ch.6

§6.7(i) Exponential Integrals
§6.7(ii) Sine and Cosine Integrals
§6.7(iii) Auxiliary Functions
§6.7(iv) Compendia

§6.7(i) Exponential Integrals

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Keywords:: exponential integrals, integral representations
Notes:: (6.7.1) and (6.7.2) follow from the definitions (§6.2(i)). (6.7.3)–(6.7.6) follow from differentiation with respect to $x$ . (6.7.7) and (6.7.8) follow from replacing the trigonometric functions by exponentials.
Permalink:: http://dlmf.nist.gov/6.7.i
See also:: Annotations for §6.7 and Ch.6

6.7.1		$\int_{0}^{\infty}\frac{e^{-at}}{t+b}\,\mathrm{d}t=\int_{0}^{\infty}\frac{e^{% iat}}{t+ib}\,\mathrm{d}t=e^{ab}E_{1}\left(ab\right),$
$a>0$ , $b>0$ ,
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $E_{1}\left(\NVar{z}\right)$ : exponential integral, $\mathrm{i}$ : imaginary unit and $\int$ : integral A&S Ref: 5.1.28 (modified form of) 5.1.29 (modified form of) Referenced by: §6.7(i) Permalink: http://dlmf.nist.gov/6.7.E1 Encodings: TeX, pMML, png See also: Annotations for §6.7(i), §6.7 and Ch.6

6.7.2		$e^{x}\int_{0}^{\alpha}\frac{e^{-xt}}{1-t}\,\mathrm{d}t=\operatorname{Ei}\left(% x\right)-\operatorname{Ei}\left((1-\alpha)x\right),$
$0\leq\alpha<1$ , $x>0$ .
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\operatorname{Ei}\left(\NVar{x}\right)$ : exponential integral, $\int$ : integral and $x$ : real variable Referenced by: §6.7(i) Permalink: http://dlmf.nist.gov/6.7.E2 Encodings: TeX, pMML, png See also: Annotations for §6.7(i), §6.7 and Ch.6

6.7.3		$\int_{x}^{\infty}\frac{e^{it}}{a^{2}+t^{2}}\,\mathrm{d}t=\frac{i}{2a}\left(e^{% a}E_{1}\left(a-ix\right)-e^{-a}E_{1}\left(-a-ix\right)\right),$
$a>0$ , $x>0$ ,
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $E_{1}\left(\NVar{z}\right)$ : exponential integral, $\mathrm{i}$ : imaginary unit, $\int$ : integral and $x$ : real variable A&S Ref: 5.1.41 (modified form of) Referenced by: §6.7(i) Permalink: http://dlmf.nist.gov/6.7.E3 Encodings: TeX, pMML, png See also: Annotations for §6.7(i), §6.7 and Ch.6

6.7.4		$\int_{x}^{\infty}\frac{te^{it}}{a^{2}+t^{2}}\,\mathrm{d}t=\tfrac{1}{2}\left(e^% {a}E_{1}\left(a-ix\right)+e^{-a}E_{1}\left(-a-ix\right)\right),$
$a>0$ , $x>0$ .
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $E_{1}\left(\NVar{z}\right)$ : exponential integral, $\mathrm{i}$ : imaginary unit, $\int$ : integral and $x$ : real variable A&S Ref: 5.1.42 (modified form of) Permalink: http://dlmf.nist.gov/6.7.E4 Encodings: TeX, pMML, png See also: Annotations for §6.7(i), §6.7 and Ch.6

6.7.5		$\int_{x}^{\infty}\frac{e^{-t}}{a^{2}+t^{2}}\,\mathrm{d}t=-\frac{1}{2ai}\left(e% ^{ia}E_{1}\left(x+ia\right)-e^{-ia}E_{1}\left(x-ia\right)\right),$
$a>0$ , $x\in\mathbb{R}$ ,
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $E_{1}\left(\NVar{z}\right)$ : exponential integral, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\mathbb{R}$ : real line and $x$ : real variable A&S Ref: 5.1.43 (modified form of) Permalink: http://dlmf.nist.gov/6.7.E5 Encodings: TeX, pMML, png See also: Annotations for §6.7(i), §6.7 and Ch.6

6.7.6		$\int_{x}^{\infty}\frac{te^{-t}}{a^{2}+t^{2}}\,\mathrm{d}t=\tfrac{1}{2}\left(e^% {ia}E_{1}\left(x+ia\right)+e^{-ia}E_{1}\left(x-ia\right)\right),$
$a>0$ , $x\in\mathbb{R}$ .
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $E_{1}\left(\NVar{z}\right)$ : exponential integral, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\mathbb{R}$ : real line and $x$ : real variable A&S Ref: 5.1.44 (modified form of) Referenced by: §6.7(i) Permalink: http://dlmf.nist.gov/6.7.E6 Encodings: TeX, pMML, png See also: Annotations for §6.7(i), §6.7 and Ch.6

6.7.7		$\int_{0}^{1}\frac{e^{-at}\sin\left(bt\right)}{t}\,\mathrm{d}t=\Im\operatorname% {Ein}\left(a+ib\right),$
$a,b\in\mathbb{R}$ ,
ⓘ Symbols: $\operatorname{Ein}\left(\NVar{z}\right)$ : complementary exponential integral, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $\Im$ : imaginary part, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\mathbb{R}$ : real line and $\sin\NVar{z}$ : sine function A&S Ref: 5.1.36 (modified form of) Referenced by: §6.7(i) Permalink: http://dlmf.nist.gov/6.7.E7 Encodings: TeX, pMML, png See also: Annotations for §6.7(i), §6.7 and Ch.6

6.7.8		$\int_{0}^{1}\frac{e^{-at}(1-\cos\left(bt\right))}{t}\,\mathrm{d}t=\Re% \operatorname{Ein}\left(a+ib\right)-\operatorname{Ein}\left(a\right),$
$a,b\in\mathbb{R}$ .
ⓘ Symbols: $\operatorname{Ein}\left(\NVar{z}\right)$ : complementary exponential integral, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part and $\mathbb{R}$ : real line A&S Ref: 5.1.38 (modified form of) Referenced by: §6.7(i) Permalink: http://dlmf.nist.gov/6.7.E8 Encodings: TeX, pMML, png See also: Annotations for §6.7(i), §6.7 and Ch.6

Many integrals with exponentials and rational functions, for example, integrals of the type $\int e^{z}R(z)\,\mathrm{d}z$ , where $R(z)$ is an arbitrary rational function, can be represented in finite form in terms of the function $E_{1}\left(z\right)$ and elementary functions; see Lebedev (1965, p. 42).

§6.7(ii) Sine and Cosine Integrals

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Keywords:: cosine integrals, integral representations, sine integrals
Notes:: See Nielsen (1906b, p. 13: there are sign errors in Eq. (27)).
Permalink:: http://dlmf.nist.gov/6.7.ii
See also:: Annotations for §6.7 and Ch.6

When $z\in\mathbb{C}$

6.7.9		$\operatorname{si}\left(z\right)=-\int_{0}^{\pi/2}e^{-z\cos t}\cos\left(z\sin t% \right)\,\mathrm{d}t,$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\operatorname{si}\left(\NVar{z}\right)$ : sine integral, $\sin\NVar{z}$ : sine function and $z$ : complex variable A&S Ref: 5.2.10 Permalink: http://dlmf.nist.gov/6.7.E9 Encodings: TeX, pMML, png See also: Annotations for §6.7(ii), §6.7 and Ch.6

6.7.10		$\operatorname{Ein}\left(z\right)-\operatorname{Cin}\left(z\right)=\int_{0}^{% \pi/2}e^{-z\cos t}\sin\left(z\sin t\right)\,\mathrm{d}t,$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\operatorname{Ein}\left(\NVar{z}\right)$ : complementary exponential integral, $\cos\NVar{z}$ : cosine function, $\operatorname{Cin}\left(\NVar{z}\right)$ : cosine integral, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\sin\NVar{z}$ : sine function and $z$ : complex variable A&S Ref: 5.2.11 (modified form of) Permalink: http://dlmf.nist.gov/6.7.E10 Encodings: TeX, pMML, png See also: Annotations for §6.7(ii), §6.7 and Ch.6

6.7.11		$\int_{0}^{1}\frac{(1-e^{-at})\cos\left(bt\right)}{t}\,\mathrm{d}t=\Re% \operatorname{Ein}\left(a+ib\right)-\operatorname{Cin}\left(b\right),$
$a,b\in\mathbb{R}$ .
ⓘ Symbols: $\operatorname{Ein}\left(\NVar{z}\right)$ : complementary exponential integral, $\cos\NVar{z}$ : cosine function, $\operatorname{Cin}\left(\NVar{z}\right)$ : cosine integral, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\in$ : element of, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\Re$ : real part and $\mathbb{R}$ : real line A&S Ref: 5.2.33 (modified form of) Permalink: http://dlmf.nist.gov/6.7.E11 Encodings: TeX, pMML, png See also: Annotations for §6.7(ii), §6.7 and Ch.6

§6.7(iii) Auxiliary Functions

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Keywords:: auxiliary functions for sine and cosine integrals, integral representations
Notes:: (6.7.12)–(6.7.14) follow from (6.5.7), (6.2.1), and (6.2.2); for the second equations in (6.7.13) and (6.7.14) see Temme (1996b, pp. 187–188). For (6.7.15) and (6.7.16) use (10.32.10).
Permalink:: http://dlmf.nist.gov/6.7.iii
See also:: Annotations for §6.7 and Ch.6

6.7.12		$\mathrm{g}\left(z\right)+i\mathrm{f}\left(z\right)=e^{-iz}\int_{z}^{\infty}% \frac{e^{it}}{t}\,\mathrm{d}t,$
$\|\operatorname{ph}z\|\leq\pi.$
ⓘ Symbols: $\pi$ : the ratio of the circumference of a circle to its diameter, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\int$ : integral, $\operatorname{ph}$ : phase, $\mathrm{f}\left(\NVar{z}\right)$ : auxiliary function for sine and cosine integrals, $\mathrm{g}\left(\NVar{z}\right)$ : auxiliary function for sine and cosine integrals and $z$ : complex variable Referenced by: §6.7(iii) Permalink: http://dlmf.nist.gov/6.7.E12 Encodings: TeX, pMML, png See also: Annotations for §6.7(iii), §6.7 and Ch.6

The path of integration does not cross the negative real axis or pass through the origin.

6.7.13	$\displaystyle\mathrm{f}\left(z\right)$	$\displaystyle=\int_{0}^{\infty}\frac{\sin t}{t+z}\,\mathrm{d}t=\int_{0}^{% \infty}\frac{e^{-zt}}{t^{2}+1}\,\mathrm{d}t,$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\sin\NVar{z}$ : sine function, $\mathrm{f}\left(\NVar{z}\right)$ : auxiliary function for sine and cosine integrals and $z$ : complex variable Referenced by: §6.12(ii), §6.7(iii) Permalink: http://dlmf.nist.gov/6.7.E13 Encodings: TeX, pMML, png See also: Annotations for §6.7(iii), §6.7 and Ch.6
6.7.14	$\displaystyle\mathrm{g}\left(z\right)$	$\displaystyle=\int_{0}^{\infty}\frac{\cos t}{t+z}\,\mathrm{d}t=\int_{0}^{% \infty}\frac{te^{-zt}}{t^{2}+1}\,\mathrm{d}t.$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\mathrm{e}$ : base of natural logarithm, $\int$ : integral, $\mathrm{g}\left(\NVar{z}\right)$ : auxiliary function for sine and cosine integrals and $z$ : complex variable A&S Ref: 5.2.13 Referenced by: §6.12(ii), §6.7(iii), §6.7(iii) Permalink: http://dlmf.nist.gov/6.7.E14 Encodings: TeX, pMML, png See also: Annotations for §6.7(iii), §6.7 and Ch.6

The first integrals on the right-hand sides apply when $|\operatorname{ph}z|<\pi$ ; the second ones when $\Re z\geq 0$ and (in the case of (6.7.14)) $z\neq 0$ .

When $|\operatorname{ph}z|<\pi$

6.7.15	$\displaystyle\mathrm{f}\left(z\right)$	$\displaystyle=2\int_{0}^{\infty}K_{0}\left(2\sqrt{zt}\right)\cos t\,\mathrm{d}t,$
ⓘ Symbols: $\cos\NVar{z}$ : cosine function, $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\mathrm{f}\left(\NVar{z}\right)$ : auxiliary function for sine and cosine integrals and $z$ : complex variable Referenced by: §6.7(iii) Permalink: http://dlmf.nist.gov/6.7.E15 Encodings: TeX, pMML, png See also: Annotations for §6.7(iii), §6.7 and Ch.6
6.7.16	$\displaystyle\mathrm{g}\left(z\right)$	$\displaystyle=2\int_{0}^{\infty}K_{0}\left(2\sqrt{zt}\right)\sin t\,\mathrm{d}t.$
ⓘ Symbols: $\,\mathrm{d}\NVar{x}$ : differential of $x$ , $\int$ : integral, $K_{\NVar{\nu}}\left(\NVar{z}\right)$ : modified Bessel function of the second kind, $\sin\NVar{z}$ : sine function, $\mathrm{g}\left(\NVar{z}\right)$ : auxiliary function for sine and cosine integrals and $z$ : complex variable Referenced by: §6.7(iii) Permalink: http://dlmf.nist.gov/6.7.E16 Encodings: TeX, pMML, png See also: Annotations for §6.7(iii), §6.7 and Ch.6

For $K_{0}$ see §10.25(ii).

§6.7(iv) Compendia

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Permalink:: http://dlmf.nist.gov/6.7.iv
See also:: Annotations for §6.7 and Ch.6

For collections of integral representations see Bierens de Haan (1939, pp. 56–59, 72–73, 82–84, 121, 133–136, 155, 179–181, 223, 225–227, 230, 259–260, 374, 377, 397–398, 408, 416, 424, 431, 438–439, 442–444, 488, 496–500, 567–571, 585, 602, 638, 675–677), Corrington (1961), Erdélyi et al. (1954a, vol. 1, pp. 267–270), Geller and Ng (1969), Nielsen (1906b), Oberhettinger (1974, pp. 244–246), Oberhettinger and Badii (1973, pp. 364–371), and Watrasiewicz (1967).

6.6 Power Series 6.8 Inequalities

NIST

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§6.7 Integral Representations

Contents

§6.7(i) Exponential Integrals

§6.7(ii) Sine and Cosine Integrals

§6.7(iii) Auxiliary Functions

§6.7(iv) Compendia