# exponential integrals

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##### 1: 8.19 Generalized Exponential Integral
###### §8.19 Generalized ExponentialIntegral
For an extensive treatment of $E_{1}\left(z\right)$ see Chapter 6. …
###### §8.19(ii) Graphics Figure 8.19.5: E 2 ⁡ ( x + i ⁢ y ) , − 3 ≤ x ≤ 3 , − 3 ≤ y ≤ 3 . … Magnify 3D Help
##### 2: 6.2 Definitions and Interrelations
###### §6.2(i) Exponential and Logarithmic Integrals
The principal value of the exponential integral $E_{1}\left(z\right)$ is defined by … Unless indicated otherwise, it is assumed throughout the DLMF that $E_{1}\left(z\right)$ assumes its principal value. …$\operatorname{Ein}\left(z\right)$ is sometimes called the complementary exponential integral. …
##### 3: 6.5 Further Interrelations
###### §6.5 Further Interrelations
6.5.2 $\operatorname{Ei}\left(x\right)=-\tfrac{1}{2}(E_{1}\left(-x+i0\right)+E_{1}% \left(-x-i0\right)),$
6.5.6 $\operatorname{Ci}\left(z\right)=-\tfrac{1}{2}(E_{1}\left(iz\right)+E_{1}\left(% -iz\right)),$
##### 4: 6.3 Graphics Figure 6.3.1: The exponential integrals E 1 ⁡ ( x ) and Ei ⁡ ( x ) , 0 < x ≤ 2 . Magnify Figure 6.3.3: | E 1 ⁡ ( x + i ⁢ y ) | , − 4 ≤ x ≤ 4 , − 4 ≤ y ≤ 4 . …Also, | E 1 ⁡ ( z ) | → ∞ logarithmically as z → 0 . Magnify 3D Help
##### 5: 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.

• Zhang and Jin (1996, pp. 652, 689) includes $\operatorname{Si}\left(x\right)$, $\operatorname{Ci}\left(x\right)$, $x=0(.5)20(2)30$, 8D; $\operatorname{Ei}\left(x\right)$, $E_{1}\left(x\right)$, $x=[0,100]$, 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 $\operatorname{Ei}\left(x\right)$, $E_{1}\left(z\right)$, 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 $\operatorname{Cin}\left(z\right)$. …
##### 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 $\operatorname{Ei}\left(x\right)$, with accuracies up to 20S.

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

• ##### 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
6.4.1 $E_{1}\left(z\right)=\operatorname{Ein}\left(z\right)-\operatorname{Ln}z-\gamma;$
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 $E_{1}\left(z\right)=\cfrac{e^{-z}}{z+\cfrac{1}{1+\cfrac{1}{z+\cfrac{2}{1+% \cfrac{2}{z+\cfrac{3}{1+\cfrac{3}{z+}}}}}}}\cdots,$ $|\operatorname{ph}z|<\pi$.