# generalized exponential integral

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##### 1: 8.19 Generalized Exponential Integral
###### §8.19 GeneralizedExponentialIntegral Figure 8.19.5: E 2 ⁡ ( x + i ⁢ y ) , - 3 ≤ x ≤ 3 , - 3 ≤ y ≤ 3 . … Magnify 3D Help
##### 2: 8.24 Physical Applications
###### §8.24(iii) GeneralizedExponentialIntegral
With more general values of $p$, $E_{p}\left(x\right)$ supplies fundamental auxiliary functions that are used in the computation of molecular electronic integrals in quantum chemistry (Harris (2002), Shavitt (1963)), and also wave acoustics of overlapping sound beams (Ding (2000)).
##### 3: 8.27 Approximations
###### §8.27(ii) GeneralizedExponentialIntegral
• Luke (1975, p. 103) gives Chebyshev-series expansions for $E_{1}\left(x\right)$ and related functions for $x\geq 5$.

##### 5: 8.20 Asymptotic Expansions of $E_{p}\left(z\right)$
###### §8.20(i) Large $z$
8.20.1 $E_{p}\left(z\right)=\frac{e^{-z}}{z}\left(\sum_{k=0}^{n-1}(-1)^{k}\frac{{\left% (p\right)_{k}}}{z^{k}}+(-1)^{n}\frac{{\left(p\right)_{n}}e^{z}}{z^{n-1}}E_{n+p% }\left(z\right)\right),$ $n=1,2,3,\dots$.
8.20.2 $E_{p}\left(z\right)\sim\frac{e^{-z}}{z}\sum_{k=0}^{\infty}(-1)^{k}\frac{{\left% (p\right)_{k}}}{z^{k}},$ $|\operatorname{ph}z|\leq\frac{3}{2}\pi-\delta$,
Where the sectors of validity of (8.20.2) and (8.20.3) overlap the contribution of the first term on the right-hand side of (8.20.3) is exponentially small compared to the other contribution; compare §2.11(ii). …
##### 6: 8.1 Special Notation
Unless otherwise indicated, primes denote derivatives with respect to the argument. The functions treated in this chapter are the incomplete gamma functions $\gamma\left(a,z\right)$, $\Gamma\left(a,z\right)$, $\gamma^{*}\left(a,z\right)$, $P\left(a,z\right)$, and $Q\left(a,z\right)$; the incomplete beta functions $\mathrm{B}_{x}\left(a,b\right)$ and $I_{x}\left(a,b\right)$; the generalized exponential integral $E_{p}\left(z\right)$; the generalized sine and cosine integrals $\mathrm{si}\left(a,z\right)$, $\mathrm{ci}\left(a,z\right)$, $\mathrm{Si}\left(a,z\right)$, and $\mathrm{Ci}\left(a,z\right)$. …
##### 8: 8.22 Mathematical Applications
###### §8.22 Mathematical Applications
8.22.1 $F_{p}\left(z\right)=\frac{\Gamma\left(p\right)}{2\pi}z^{1-p}E_{p}\left(z\right% )=\frac{\Gamma\left(p\right)}{2\pi}\Gamma\left(1-p,z\right),$
##### 10: 2.11 Remainder Terms; Stokes Phenomenon
From §8.19(i) the generalized exponential integral is given by However, on combining (2.11.6) with the connection formula (8.19.18), with $m=1$, we derive … Two different asymptotic expansions in terms of elementary functions, (2.11.6) and (2.11.7), are available for the generalized exponential integral in the sector $\frac{1}{2}\pi<\operatorname{ph}z<\frac{3}{2}\pi$. …