§6.7 Integral Representations

Keywords:: exponential integrals
Permalink:: http://dlmf.nist.gov/6.7

§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

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.
Keywords:: exponential integrals
Permalink:: http://dlmf.nist.gov/6.7.i

6.7.1 $\int _{0}^{\infty}\frac{e^{{-at}}}{t+b}dt=\int _{0}^{\infty}\frac{e^{{iat}}}{t+ib}dt=e^{{ab}}\mathop{E_{1}\/}\nolimits\!\left(ab\right),$ $a>0$ , $b>0$ ,

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ : exponential integral 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
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6.7.2 $e^{x}\int _{0}^{{\alpha}}\frac{e^{{-xt}}}{1-t}dt=\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)-\mathop{\mathrm{Ei}\/}\nolimits\!\left((1-\alpha)x\right),$ $0\leq\alpha<1$ , $x>0$ .

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)$ : exponential integral, $\int$ : integral and $x$ : real variable
Referenced by:: §6.7(i)
Permalink:: http://dlmf.nist.gov/6.7.E2
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6.7.3 $\int _{x}^{\infty}\frac{e^{{it}}}{a^{2}+t^{2}}dt=\frac{i}{2a}\left(e^{a}\mathop{E_{1}\/}\nolimits\!\left(a-ix\right)-e^{{-a}}\mathop{E_{1}\/}\nolimits\!\left(-a-ix\right)\right),$ $a>0$ , $x>0$ ,

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ : exponential integral, $\int$ : 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
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6.7.4 $\int _{x}^{\infty}\frac{te^{{it}}}{a^{2}+t^{2}}dt=\tfrac{1}{2}\left(e^{a}\mathop{E_{1}\/}\nolimits\!\left(a-ix\right)+e^{{-a}}\mathop{E_{1}\/}\nolimits\!\left(-a-ix\right)\right),$ $a>0$ , $x>0$ .

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ : exponential integral, $\int$ : integral and $x$ : real variable
A&S Ref:: 5.1.42 (modified form of)
Permalink:: http://dlmf.nist.gov/6.7.E4
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6.7.5 $\int _{x}^{{\infty}}\frac{e^{{-t}}}{a^{2}+t^{2}}dt=-\frac{1}{2ai}\left(e^{{ia}}\mathop{E_{1}\/}\nolimits\!\left(x+ia\right)-e^{{-ia}}\mathop{E_{1}\/}\nolimits\!\left(x-ia\right)\right),$ $a>0$ , $x\in\Real$ ,

Symbols:: $dx$ : differential of $x$ , $\in$ : element of, $e$ : base of exponential function, $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ : exponential integral, $\int$ : integral, $\Real$ : real line (excluding infinity) and $x$ : real variable
A&S Ref:: 5.1.43 (modified form of)
Permalink:: http://dlmf.nist.gov/6.7.E5
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6.7.6 $\int _{x}^{{\infty}}\frac{te^{{-t}}}{a^{2}+t^{2}}dt=\tfrac{1}{2}\left(e^{{ia}}\mathop{E_{1}\/}\nolimits\!\left(x+ia\right)+e^{{-ia}}\mathop{E_{1}\/}\nolimits\!\left(x-ia\right)\right),$ $a>0$ , $x\in\Real$ .

Symbols:: $dx$ : differential of $x$ , $\in$ : element of, $e$ : base of exponential function, $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ : exponential integral, $\int$ : integral, $\Real$ : real line (excluding infinity) 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
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6.7.7 $\int _{0}^{1}\frac{e^{{-at}}\mathop{\sin\/}\nolimits\!\left(bt\right)}{t}dt=\imagpart{\mathop{\mathrm{Ein}\/}\nolimits\!\left(a+ib\right)},$ $a,b\in\Real$ ,

Symbols:: $\mathop{\mathrm{Ein}\/}\nolimits\!\left(z\right)$ : complementary exponential integral, $dx$ : differential of $x$ , $\in$ : element of, $e$ : base of exponential function, $\imagpart{}$ : imaginary part, $\int$ : integral, $\Real$ : real line (excluding infinity) and $\mathop{\sin\/}\nolimits z$ : sine function
A&S Ref:: 5.1.36 (modified form of)
Referenced by:: §6.7(i)
Permalink:: http://dlmf.nist.gov/6.7.E7
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6.7.8 $\int _{0}^{1}\frac{e^{{-at}}(1-\mathop{\cos\/}\nolimits\!\left(bt\right))}{t}dt=\realpart{\mathop{\mathrm{Ein}\/}\nolimits\!\left(a+ib\right)}-\mathop{\mathrm{Ein}\/}\nolimits\!\left(a\right),$ $a,b\in\Real$ .

Symbols:: $\mathop{\mathrm{Ein}\/}\nolimits\!\left(z\right)$ : complementary exponential integral, $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\in$ : element of, $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : real part and $\Real$ : real line (excluding infinity)
A&S Ref:: 5.1.38 (modified form of)
Referenced by:: §6.7(i)
Permalink:: http://dlmf.nist.gov/6.7.E8
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Many integrals with exponentials and rational functions, for example, integrals of the type $\int e^{z}R(z)dz$ , where $R(z)$ is an arbitrary rational function, can be represented in finite form in terms of the function $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ and elementary functions; see Lebedev (1965, p. 42).

§6.7(ii) Sine and Cosine Integrals

Notes:: See Nielsen (1906b, p. 13: there are sign errors in Eq. (27)).
Keywords:: cosine integrals, sine integrals
Permalink:: http://dlmf.nist.gov/6.7.ii

When $z\in\Complex$

6.7.9 $\mathop{\mathrm{si}\/}\nolimits\!\left(z\right)=-\int _{0}^{{\pi/2}}e^{{-z\mathop{\cos\/}\nolimits t}}\mathop{\cos\/}\nolimits\!\left(z\mathop{\sin\/}\nolimits t\right)dt,$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\mathrm{si}\/}\nolimits\!\left(z\right)$ : sine integral, $\mathop{\sin\/}\nolimits z$ : sine function and $z$ : complex variable
A&S Ref:: 5.2.10
Permalink:: http://dlmf.nist.gov/6.7.E9
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6.7.10 $\mathop{\mathrm{Ein}\/}\nolimits\!\left(z\right)-\mathop{\mathrm{Cin}\/}\nolimits\!\left(z\right)=\int _{0}^{{\pi/2}}e^{{-z\mathop{\cos\/}\nolimits t}}\mathop{\sin\/}\nolimits\!\left(z\mathop{\sin\/}\nolimits t\right)dt,$

Symbols:: $\mathop{\mathrm{Ein}\/}\nolimits\!\left(z\right)$ : complementary exponential integral, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\mathrm{Cin}\/}\nolimits\!\left(z\right)$ : cosine integral, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function and $z$ : complex variable
A&S Ref:: 5.2.11 (modified form of)
Permalink:: http://dlmf.nist.gov/6.7.E10
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6.7.11 $\int _{0}^{1}\frac{(1-e^{{-at}})\mathop{\cos\/}\nolimits\!\left(bt\right)}{t}dt=\realpart{\mathop{\mathrm{Ein}\/}\nolimits\!\left(a+ib\right)}-\mathop{\mathrm{Cin}\/}\nolimits\!\left(b\right),$ $a,b\in\Real$ .

Symbols:: $\mathop{\mathrm{Ein}\/}\nolimits\!\left(z\right)$ : complementary exponential integral, $\mathop{\cos\/}\nolimits z$ : cosine function, $\mathop{\mathrm{Cin}\/}\nolimits\!\left(z\right)$ : cosine integral, $dx$ : differential of $x$ , $\in$ : element of, $e$ : base of exponential function, $\int$ : integral, $\realpart{}$ : real part and $\Real$ : real line (excluding infinity)
A&S Ref:: 5.2.33 (modified form of)
Permalink:: http://dlmf.nist.gov/6.7.E11
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§6.7(iii) Auxiliary Functions

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).
Keywords:: auxiliary functions for sine and cosine integrals
Permalink:: http://dlmf.nist.gov/6.7.iii

6.7.12 $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)+i\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)=e^{{-iz}}\int _{z}^{\infty}\frac{e^{{it}}}{t}dt,$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi.$

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)$ : auxiliary function for sine and cosine integrals, $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)$ : auxiliary function for sine and cosine integrals and $z$ : complex variable
Referenced by:: §6.7(iii)
Permalink:: http://dlmf.nist.gov/6.7.E12
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The path of integration does not cross the negative real axis or pass through the origin.

6.7.13 $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)=\int _{0}^{\infty}\frac{\mathop{\sin\/}\nolimits t}{t+z}dt=\int _{0}^{\infty}\frac{e^{{-zt}}}{t^{2}+1}dt,$

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)$ : 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
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6.7.14 $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)=\int _{0}^{\infty}\frac{\mathop{\cos\/}\nolimits t}{t+z}dt=\int _{0}^{\infty}\frac{te^{{-zt}}}{t^{2}+1}dt.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $e$ : base of exponential function, $\int$ : integral, $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)$ : 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
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The first integrals on the right-hand sides apply when $|\mathop{\mathrm{ph}\/}\nolimits z|<\pi$ ; the second ones when $\realpart{z}\geq 0$ and (in the case of (6.7.14)) $z\neq 0$ .

When $|\mathop{\mathrm{ph}\/}\nolimits z|<\pi$

6.7.15 $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)=2\int _{0}^{\infty}\mathop{K_{{0}}\/}\nolimits\!\left(2\sqrt{zt}\right)\mathop{\cos\/}\nolimits tdt,$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $\int$ : integral, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)$ : auxiliary function for sine and cosine integrals and $z$ : complex variable
Referenced by:: §6.7(iii)
Permalink:: http://dlmf.nist.gov/6.7.E15
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6.7.16 $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)=2\int _{0}^{\infty}\mathop{K_{{0}}\/}\nolimits\!\left(2\sqrt{zt}\right)\mathop{\sin\/}\nolimits tdt.$

Symbols:: $dx$ : differential of $x$ , $\int$ : integral, $\mathop{K_{{\nu}}\/}\nolimits\!\left(z\right)$ : modified Bessel function, $\mathop{\sin\/}\nolimits z$ : sine function, $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)$ : auxiliary function for sine and cosine integrals and $z$ : complex variable
Referenced by:: §6.7(iii)
Permalink:: http://dlmf.nist.gov/6.7.E16
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For $\mathop{K_{{0}}\/}\nolimits$ see §10.25(ii).

§6.7(iv) Compendia

Permalink:: http://dlmf.nist.gov/6.7.iv

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).

§6.7 Integral Representations

Contents

§6.7(i) Exponential Integrals

§6.7(ii) Sine and Cosine Integrals

§6.7(iii) Auxiliary Functions

§6.7(iv) Compendia