§6.7 Integral Representations

§6.7(i) Exponential Integrals

 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 $e^{x}\int_{0}^{\alpha}\frac{e^{-xt}}{1-t}\,\mathrm{d}t=\operatorname{Ei}\left(% x\right)-\operatorname{Ei}\left((1-\alpha)x\right),$ $0\leq\alpha<1$, $x>0$.
 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 $\int_{x}^{\infty}\frac{e^{-t}}{a^{2}+t^{2}}\,\mathrm{d}t=-\frac{1}{2ai}\left(e% ^{ia}E_{1}\left(x+ia\right)-e^{-ia}E_{1}\left(x-ia\right)\right),$ $a>0$, $x\in\mathbb{R}$,
 6.7.6 $\int_{x}^{\infty}\frac{te^{-t}}{a^{2}+t^{2}}\,\mathrm{d}t=\tfrac{1}{2}\left(e^% {ia}E_{1}\left(x+ia\right)+e^{-ia}E_{1}\left(x-ia\right)\right),$ $a>0$, $x\in\mathbb{R}$.
 6.7.7 $\int_{0}^{1}\frac{e^{-at}\sin\left(bt\right)}{t}\,\mathrm{d}t=\Im\operatorname% {Ein}\left(a+ib\right),$ $a,b\in\mathbb{R}$,
 6.7.8 $\int_{0}^{1}\frac{e^{-at}(1-\cos\left(bt\right))}{t}\,\mathrm{d}t=\Re% \operatorname{Ein}\left(a+ib\right)-\operatorname{Ein}\left(a\right),$ $a,b\in\mathbb{R}$.

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

When $z\in\mathbb{C}$

 6.7.9 $\operatorname{si}\left(z\right)=-\int_{0}^{\pi/2}e^{-z\cos t}\cos\left(z\sin t% \right)\,\mathrm{d}t,$
 6.7.10 $\operatorname{Ein}\left(z\right)-\operatorname{Cin}\left(z\right)=\int_{0}^{% \pi/2}e^{-z\cos t}\sin\left(z\sin t\right)\,\mathrm{d}t,$
 6.7.11 $\int_{0}^{1}\frac{(1-e^{-at})\cos\left(bt\right)}{t}\,\mathrm{d}t=\Re% \operatorname{Ein}\left(a+ib\right)-\operatorname{Cin}\left(b\right),$ $a,b\in\mathbb{R}$.

§6.7(iii) Auxiliary Functions

 6.7.12 $\mathrm{g}\left(z\right)+i\mathrm{f}\left(z\right)=e^{-iz}\int_{z}^{\infty}% \frac{e^{it}}{t}\,\mathrm{d}t,$ $|\operatorname{ph}z|\leq\pi.$

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

 6.7.13 $\displaystyle\mathrm{f}\left(z\right)$ $\displaystyle=\int_{0}^{\infty}\frac{\sin t}{t+z}\,\mathrm{d}t=\int_{0}^{% \infty}\frac{e^{-zt}}{t^{2}+1}\,\mathrm{d}t,$ 6.7.14 $\displaystyle\mathrm{g}\left(z\right)$ $\displaystyle=\int_{0}^{\infty}\frac{\cos t}{t+z}\,\mathrm{d}t=\int_{0}^{% \infty}\frac{te^{-zt}}{t^{2}+1}\,\mathrm{d}t.$

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 $\displaystyle\mathrm{f}\left(z\right)$ $\displaystyle=2\int_{0}^{\infty}K_{0}\left(2\sqrt{zt}\right)\cos t\,\mathrm{d}t,$ 6.7.16 $\displaystyle\mathrm{g}\left(z\right)$ $\displaystyle=2\int_{0}^{\infty}K_{0}\left(2\sqrt{zt}\right)\sin t\,\mathrm{d}t.$

For $K_{0}$ see §10.25(ii).

§6.7(iv) Compendia

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).