# complementary exponential integral

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##### 1: 6.1 Special Notation
The main functions treated in this chapter are the exponential integrals $\operatorname{Ei}\left(x\right)$, $E_{1}\left(z\right)$, and $\operatorname{Ein}\left(z\right)$; the logarithmic integral $\operatorname{li}\left(x\right)$; the sine integrals $\operatorname{Si}\left(z\right)$ and $\operatorname{si}\left(z\right)$; the cosine integrals $\operatorname{Ci}\left(z\right)$ and $\operatorname{Cin}\left(z\right)$. …
##### 2: 6.20 Approximations
• Luke (1969b, pp. 41–42) gives Chebyshev expansions of $\operatorname{Ein}\left(ax\right)$, $\operatorname{Si}\left(ax\right)$, and $\operatorname{Cin}\left(ax\right)$ for $-1\leq x\leq 1$, $a\in\mathbb{C}$. The coefficients are given in terms of series of Bessel functions.

• Luke (1969b, pp. 321–322) covers $\operatorname{Ein}\left(x\right)$ and $-\operatorname{Ein}\left(-x\right)$ for $0\leq x\leq 8$ (the Chebyshev coefficients are given to 20D); $E_{1}\left(x\right)$ for $x\geq 5$ (20D), and $\operatorname{Ei}\left(x\right)$ for $x\geq 8$ (15D). Coefficients for the sine and cosine integrals are given on pp. 325–327.

• Luke (1969b, pp. 402, 410, and 415–421) gives main diagonal Padé approximations for $\operatorname{Ein}\left(z\right)$, $\operatorname{Si}\left(z\right)$, $\operatorname{Cin}\left(z\right)$ (valid near the origin), and $E_{1}\left(z\right)$ (valid for large $|z|$); approximate errors are given for a selection of $z$-values.

• Luke (1969b, pp. 411–414) gives rational approximations for $\operatorname{Ein}\left(z\right)$.

• ##### 3: 6.4 Analytic Continuation
6.4.1 $E_{1}\left(z\right)=\operatorname{Ein}\left(z\right)-\operatorname{Ln}z-\gamma;$
6.4.3 $E_{1}\left(ze^{\pm\pi i}\right)=\operatorname{Ein}\left(-z\right)-\ln z-\gamma% \mp\pi i,$ $|\operatorname{ph}z|\leq\pi$.
##### 4: 6.7 Integral Representations
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 $\int_{0}^{1}\frac{e^{-at}(1-\cos\left(bt\right))}{t}\,\mathrm{d}t=\Re% \operatorname{Ein}\left(a+ib\right)-\operatorname{Ein}\left(a\right),$ $a,b\in\mathbb{R}$.
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 $\int_{0}^{1}\frac{(1-e^{-at})\cos\left(bt\right)}{t}\,\mathrm{d}t=\Re% \operatorname{Ein}\left(a+ib\right)-\operatorname{Cin}\left(b\right),$ $a,b\in\mathbb{R}$.
##### 5: 6.2 Definitions and Interrelations
6.2.3 $\operatorname{Ein}\left(z\right)=\int_{0}^{z}\frac{1-e^{-t}}{t}\,\mathrm{d}t.$
$\operatorname{Ein}\left(z\right)$ is sometimes called the complementary exponential integral. …
6.2.4 $E_{1}\left(z\right)=\operatorname{Ein}\left(z\right)-\ln z-\gamma.$
##### 7: 6.19 Tables
• Abramowitz and Stegun (1964, Chapter 5) includes $x^{-1}\operatorname{Si}\left(x\right)$, $-x^{-2}\operatorname{Cin}\left(x\right)$, $x^{-1}\operatorname{Ein}\left(x\right)$, $-x^{-1}\operatorname{Ein}\left(-x\right)$, $x=0(.01)0.5$; $\operatorname{Si}\left(x\right)$, $\operatorname{Ci}\left(x\right)$, $\operatorname{Ei}\left(x\right)$, $E_{1}\left(x\right)$, $x=0.5(.01)2$; $\operatorname{Si}\left(x\right)$, $\operatorname{Ci}\left(x\right)$, $xe^{-x}\operatorname{Ei}\left(x\right)$, $xe^{x}E_{1}\left(x\right)$, $x=2(.1)10$; $x\mathrm{f}\left(x\right)$, $x^{2}\mathrm{g}\left(x\right)$, $xe^{-x}\operatorname{Ei}\left(x\right)$, $xe^{x}E_{1}\left(x\right)$, $x^{-1}=0(.005)0.1$; $\operatorname{Si}\left(\pi x\right)$, $\operatorname{Cin}\left(\pi x\right)$, $x=0(.1)10$. Accuracy varies but is within the range 8S–11S.

• ##### 8: 6.10 Other Series Expansions
6.10.8 $\operatorname{Ein}\left(z\right)=ze^{-z/2}\left({\mathsf{i}^{(1)}_{0}}\left(% \tfrac{1}{2}z\right)+\sum_{n=1}^{\infty}\dfrac{2n+1}{n(n+1)}{\mathsf{i}^{(1)}_% {n}}\left(\tfrac{1}{2}z\right)\right).$
##### 9: 7.18 Repeated Integrals of the Complementary Error Function
$\mathop{\mathrm{i}^{-1}\mathrm{erfc}}\left(z\right)=\frac{2}{\sqrt{\pi}}e^{-z^% {2}},$
7.18.4 $\frac{{\mathrm{d}}^{n}}{{\mathrm{d}z}^{n}}\left(e^{z^{2}}\operatorname{erfc}z% \right)=(-1)^{n}2^{n}n!e^{z^{2}}\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z% \right),$ $n=0,1,2,\dots$.
7.18.9 $\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)=e^{-z^{2}}\left(\frac{1}{2^% {n}\Gamma\left(\tfrac{1}{2}n+1\right)}M\left(\tfrac{1}{2}n+\tfrac{1}{2},\tfrac% {1}{2},z^{2}\right)-\frac{z}{2^{n-1}\Gamma\left(\tfrac{1}{2}n+\tfrac{1}{2}% \right)}M\left(\tfrac{1}{2}n+1,\tfrac{3}{2},z^{2}\right)\right),$
7.18.10 $\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)=\frac{e^{-z^{2}}}{2^{n}% \sqrt{\pi}}U\left(\tfrac{1}{2}n+\tfrac{1}{2},\tfrac{1}{2},z^{2}\right).$
7.18.11 $\mathop{\mathrm{i}^{n}\mathrm{erfc}}\left(z\right)=\frac{e^{-z^{2}/2}}{\sqrt{2% ^{n-1}\pi}}U\left(n+\tfrac{1}{2},z\sqrt{2}\right).$