# exponential integrals

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##### 1: 8.19 Generalized Exponential Integral
###### §8.19 Generalized ExponentialIntegral
8.19.1 $E_{p}\left(z\right)=z^{p-1}\Gamma\left(1-p,z\right).$
For an extensive treatment of $E_{1}\left(z\right)$ see Chapter 6. …
##### 2: 6.2 Definitions and Interrelations
###### §6.2(i) Exponential and Logarithmic Integrals
6.2.1 $E_{1}\left(z\right)=\int_{z}^{\infty}\frac{e^{-t}}{t}\mathrm{d}t,$ $z\neq 0$,
6.2.3 $\mathrm{Ein}\left(z\right)=\int_{0}^{z}\frac{1-e^{-t}}{t}\mathrm{d}t.$
$\mathrm{Ein}\left(z\right)$ is sometimes called the complementary exponential integral. …
##### 3: 6.5 Further Interrelations
###### §6.5 Further Interrelations
6.5.2 $\mathrm{Ei}\left(x\right)=-\tfrac{1}{2}(E_{1}\left(-x+i0\right)+E_{1}\left(-x-% i0\right)),$
6.5.6 $\mathrm{Ci}\left(z\right)=-\tfrac{1}{2}(E_{1}\left(iz\right)+E_{1}\left(-iz% \right)),$
##### 5: 6.19 Tables
• Abramowitz and Stegun (1964, Chapter 5) includes $x^{-1}\mathrm{Si}\left(x\right)$, $-x^{-2}\mathrm{Cin}\left(x\right)$, $x^{-1}\mathrm{Ein}\left(x\right)$, $-x^{-1}\mathrm{Ein}\left(-x\right)$, $x=0(.01)0.5$; $\mathrm{Si}\left(x\right)$, $\mathrm{Ci}\left(x\right)$, $\mathrm{Ei}\left(x\right)$, $E_{1}\left(x\right)$, $x=0.5(.01)2$; $\mathrm{Si}\left(x\right)$, $\mathrm{Ci}\left(x\right)$, $xe^{-x}\mathrm{Ei}\left(x\right)$, $xe^{x}E_{1}\left(x\right)$, $x=2(.1)10$; $x\mathrm{f}\left(x\right)$, $x^{2}\mathrm{g}\left(x\right)$, $xe^{-x}\mathrm{Ei}\left(x\right)$, $xe^{x}E_{1}\left(x\right)$, $x^{-1}=0(.005)0.1$; $\mathrm{Si}\left(\pi x\right)$, $\mathrm{Cin}\left(\pi x\right)$, $x=0(.1)10$. Accuracy varies but is within the range 8S–11S.

• Zhang and Jin (1996, pp. 652, 689) includes $\mathrm{Si}\left(x\right)$, $\mathrm{Ci}\left(x\right)$, $x=0(.5)20(2)30$, 8D; $\mathrm{Ei}\left(x\right)$, $E_{1}\left(x\right)$, $x=[0,100]$, 8S.

• Abramowitz and Stegun (1964, Chapter 5) includes the real and imaginary parts of $ze^{z}E_{1}\left(z\right)$, $x=-19(1)20$, $y=0(1)20$, 6D; $e^{z}E_{1}\left(z\right)$, $x=-4(.5)-2$, $y=0(.2)1$, 6D; $E_{1}\left(z\right)+\ln z$, $x=-2(.5)2.5$, $y=0(.2)1$, 6D.

• Zhang and Jin (1996, pp. 690–692) includes the real and imaginary parts of $E_{1}\left(z\right)$, $\pm x=0.5,1,3,5,10,15,20,50,100$, $y=0(.5)1(1)5(5)30,50,100$, 8S.

• ##### 6: 6.1 Special Notation
Unless otherwise noted, primes indicate derivatives with respect to the argument. The main functions treated in this chapter are the exponential integrals $\mathrm{Ei}\left(x\right)$, $E_{1}\left(z\right)$, and $\mathrm{Ein}\left(z\right)$; the logarithmic integral $\mathrm{li}\left(x\right)$; the sine integrals $\mathrm{Si}\left(z\right)$ and $\mathrm{si}\left(z\right)$; the cosine integrals $\mathrm{Ci}\left(z\right)$ and $\mathrm{Cin}\left(z\right)$. …
##### 8: 6.20 Approximations
• Hastings (1955) gives several minimax polynomial and rational approximations for $E_{1}\left(x\right)+\ln x$, $xe^{x}E_{1}\left(x\right)$, and the auxiliary functions $\mathrm{f}\left(x\right)$ and $\mathrm{g}\left(x\right)$. These are included in Abramowitz and Stegun (1964, Ch. 5).

• Cody and Thacher (1968) provides minimax rational approximations for $E_{1}\left(x\right)$, with accuracies up to 20S.

• Cody and Thacher (1969) provides minimax rational approximations for $\mathrm{Ei}\left(x\right)$, with accuracies up to 20S.

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

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

• ##### 9: 6.4 Analytic Continuation
###### §6.4 Analytic Continuation
Analytic continuation of the principal value of $E_{1}\left(z\right)$ yields a multi-valued function with branch points at $z=0$ and $z=\infty$. The general value of $E_{1}\left(z\right)$ is given by
6.4.2 $E_{1}\left(ze^{2m\pi i}\right)=E_{1}\left(z\right)-2m\pi i,$ $m\in\mathbb{Z}$,
##### 10: 6.9 Continued Fraction
###### §6.9 Continued Fraction
6.9.1 $E_{1}\left(z\right)=\cfrac{e^{-z}}{z+\cfrac{1}{1+\cfrac{1}{z+\cfrac{2}{1+% \cfrac{2}{z+\cfrac{3}{1+\cfrac{3}{z+}}}}}}}\cdots,$ $|\operatorname{ph}z|<\pi$.