# generalized exponentials

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##### 2: 8.24 Physical Applications
###### §8.24(iii) GeneralizedExponential Integral
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: 4.44 Other Applications
###### §4.44 Other Applications
For applications of generalized exponentials and generalized logarithms to computer arithmetic see §3.1(iv). …
##### 4: 4.12 Generalized Logarithms and Exponentials
###### §4.12 Generalized Logarithms and Exponentials
A generalized exponential function $\phi(x)$ satisfies the equations …
4.12.2 $\phi(0)=0,$
4.12.5 $\phi(x)=\psi(x)=x,$ $0\leq x\leq 1$.
4.12.6 $\phi(x)=\ln\left(x+1\right),$ $-1,
##### 5: 8.27 Approximations
###### §8.27(ii) GeneralizedExponential Integral
• Luke (1975, p. 103) gives Chebyshev-series expansions for $E_{1}\left(x\right)$ and related functions for $x\geq 5$.

• ##### 6: 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). …
##### 8: 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 $\operatorname{si}\left(a,z\right)$, $\operatorname{ci}\left(a,z\right)$, $\operatorname{Si}\left(a,z\right)$, and $\operatorname{Ci}\left(a,z\right)$. …
##### 10: 8.22 Mathematical Applications
###### §8.22 Mathematical Applications
8.22.1 $F_{p}\left(z\right)=\frac{\Gamma\left(p\right)}{2\pi}z^{1-p}E_{p}\left(z\right% )=\frac{\Gamma\left(p\right)}{2\pi}\Gamma\left(1-p,z\right),$