# §8.19 Generalized Exponential Integral

## §8.19(i) Definition and Integral Representations

For $p,z\in\mathbb{C}$

 8.19.1 $E_{p}\left(z\right)=z^{p-1}\Gamma\left(1-p,z\right).$ ⓘ Defines: $E_{\NVar{p}}\left(\NVar{z}\right)$: generalized exponential integral Symbols: $\Gamma\left(\NVar{a},\NVar{z}\right)$: incomplete gamma function, $z$: complex variable and $p$: parameter A&S Ref: 5.1.45 6.5.9 (Definition extended to general values of $p$.) Referenced by: §8.19(i), §8.19(iii), §8.19(iv), §8.19(v), §8.19(vi), §8.19(vii) Permalink: http://dlmf.nist.gov/8.19.E1 Encodings: TeX, pMML, png See also: Annotations for §8.19(i), §8.19 and Ch.8

Most properties of $E_{p}\left(z\right)$ follow straightforwardly from those of $\Gamma\left(a,z\right)$. For an extensive treatment of $E_{1}\left(z\right)$ see Chapter 6.

 8.19.2 $E_{p}\left(z\right)=z^{p-1}\int_{z}^{\infty}\frac{e^{-t}}{t^{p}}\mathrm{d}t.$

When the path of integration excludes the origin and does not cross the negative real axis (8.19.2) defines the principal value of $E_{p}\left(z\right)$, and unless indicated otherwise in the DLMF principal values are assumed.

### Other Integral Representations

 8.19.3 $\displaystyle E_{p}\left(z\right)$ $\displaystyle=\int_{1}^{\infty}\frac{e^{-zt}}{t^{p}}\mathrm{d}t,$ $|\operatorname{ph}z|<\tfrac{1}{2}\pi$, 8.19.4 $\displaystyle E_{p}\left(z\right)$ $\displaystyle=\frac{z^{p-1}e^{-z}}{\Gamma\left(p\right)}\int_{0}^{\infty}\frac% {t^{p-1}e^{-zt}}{1+t}\mathrm{d}t,$ $|\operatorname{ph}z|<\tfrac{1}{2}\pi$, $\Re p>0$.

Integral representations of Mellin–Barnes type for $E_{p}\left(z\right)$ follow immediately from (8.6.11), (8.6.12), and (8.19.1).

## §8.19(ii) Graphics

In Figures 8.19.28.19.5, height corresponds to the absolute value of the function and color to the phase. See About Color Map.

## §8.19(iii) Special Values

 8.19.5 $E_{0}\left(z\right)=z^{-1}e^{-z},$ $z\neq 0$, ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $E_{\NVar{p}}\left(\NVar{z}\right)$: generalized exponential integral and $z$: complex variable A&S Ref: 5.1.24 Permalink: http://dlmf.nist.gov/8.19.E5 Encodings: TeX, pMML, png See also: Annotations for §8.19(iii), §8.19 and Ch.8
 8.19.6 $E_{p}\left(0\right)=\frac{1}{p-1},$ $\Re p>1$, ⓘ Symbols: $E_{\NVar{p}}\left(\NVar{z}\right)$: generalized exponential integral, $\Re$: real part and $p$: parameter A&S Ref: 5.1.23 (Extended to general values of $p$.) Permalink: http://dlmf.nist.gov/8.19.E6 Encodings: TeX, pMML, png See also: Annotations for §8.19(iii), §8.19 and Ch.8
 8.19.7 $E_{n}\left(z\right)=\frac{(-z)^{n-1}}{(n-1)!}E_{1}\left(z\right)+\frac{e^{-z}}% {(n-1)!}\sum_{k=0}^{n-2}(n-k-2)!(-z)^{k},$ $n=2,3,\dots$.

## §8.19(iv) Series Expansions

For $n=1,2,3,\dots$,

 8.19.8 $E_{n}\left(z\right)=\frac{(-z)^{n-1}}{(n-1)!}(\psi\left(n\right)-\ln z)-\sum_{% \begin{subarray}{c}k=0\\ k\neq n-1\end{subarray}}^{\infty}\frac{(-z)^{k}}{k!(1-n+k)},$

and

 8.19.9 $E_{n}\left(z\right)=\frac{(-1)^{n}z^{n-1}}{(n-1)!}\ln z+\frac{e^{-z}}{(n-1)!}% \sum_{k=1}^{n-1}(-z)^{k-1}\Gamma\left(n-k\right)+\frac{e^{-z}(-z)^{n-1}}{(n-1)% !}\sum_{k=0}^{\infty}\frac{z^{k}}{k!}\psi\left(k+1\right),$

with $|\operatorname{ph}z|\leq\pi$ in both equations. For $\psi\left(x\right)$ see §5.2(i).

When $p\in\mathbb{C}$

 8.19.10 $E_{p}\left(z\right)=z^{p-1}\Gamma\left(1-p\right)-\sum_{k=0}^{\infty}\frac{(-z% )^{k}}{k!(1-p+k)},$
 8.19.11 $E_{p}\left(z\right)=\Gamma\left(1-p\right)\left(z^{p-1}-e^{-z}\sum_{k=0}^{% \infty}\frac{z^{k}}{\Gamma\left(2-p+k\right)}\right),$

again with $|\operatorname{ph}z|\leq\pi$ in both equations. The right-hand sides are replaced by their limiting forms when $p=1,2,3,\dots$.

## §8.19(v) Recurrence Relation and Derivatives

 8.19.12 $pE_{p+1}\left(z\right)+zE_{p}\left(z\right)=e^{-z}.$
 8.19.13 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}E_{p}\left(z\right)$ $\displaystyle=-E_{p-1}\left(z\right),$ ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$, $E_{\NVar{p}}\left(\NVar{z}\right)$: generalized exponential integral, $z$: complex variable and $p$: parameter A&S Ref: 5.1.36 (Extended to general values of $p$.) Referenced by: §8.19(x) Permalink: http://dlmf.nist.gov/8.19.E13 Encodings: TeX, pMML, png See also: Annotations for §8.19(v), §8.19 and Ch.8 8.19.14 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}(e^{z}E_{p}\left(z\right))$ $\displaystyle=e^{z}E_{p}\left(z\right)\left(1+\frac{p-1}{z}\right)-\frac{1}{z}.$

### $p$-Derivatives

For $j=1,2,3,\dots$,

 8.19.15 $\frac{{\partial}^{j}E_{p}\left(z\right)}{{\partial p}^{j}}=(-1)^{j}\int_{1}^{% \infty}(\ln t)^{j}t^{-p}e^{-zt}\mathrm{d}t,$ $\Re z>0$.

For properties and numerical tables see Milgram (1985), and also (when $p=1$) MacLeod (2002b).

## §8.19(vi) Relation to Confluent Hypergeometric Function

 8.19.16 $E_{p}\left(z\right)=z^{p-1}e^{-z}U\left(p,p,z\right).$

For $U\left(a,b,z\right)$ see §13.2(i).

## §8.19(vii) Continued Fraction

 8.19.17 $E_{p}\left(z\right)=e^{-z}\left(\cfrac{1}{z+\cfrac{p}{1+\cfrac{1}{z+\cfrac{p+1% }{1+\cfrac{2}{z+\cdots}}}}}\right),$ $|\operatorname{ph}z|<\pi$.

See also Cuyt et al. (2008, pp. 277–285).

## §8.19(viii) Analytic Continuation

The general function $E_{p}\left(z\right)$ is attained by extending the path in (8.19.2) across the negative real axis. Unless $p$ is a nonpositive integer, $E_{p}\left(z\right)$ has a branch point at $z=0$. For $z\neq 0$ each branch of $E_{p}\left(z\right)$ is an entire function of $p$.

 8.19.18 $E_{p}\left(ze^{2m\pi i}\right)=\frac{2\pi ie^{mp\pi i}}{\Gamma\left(p\right)}% \frac{\sin\left(mp\pi\right)}{\sin\left(p\pi\right)}z^{p-1}+E_{p}\left(z\right),$ $m\in\mathbb{Z}$, $z\neq 0$.

## §8.19(ix) Inequalities

For $n=1,2,3,\dots$ and $x>0$,

 8.19.19 $\frac{n-1}{n}E_{n}\left(x\right) ⓘ Symbols: $E_{\NVar{p}}\left(\NVar{z}\right)$: generalized exponential integral, $x$: real variable and $n$: nonnegative integer A&S Ref: 5.1.17 Permalink: http://dlmf.nist.gov/8.19.E19 Encodings: TeX, pMML, png See also: Annotations for §8.19(ix), §8.19 and Ch.8
 8.19.20 $\left(E_{n}\left(x\right)\right)^{2} ⓘ Symbols: $E_{\NVar{p}}\left(\NVar{z}\right)$: generalized exponential integral, $x$: real variable and $n$: nonnegative integer A&S Ref: 5.1.18 Permalink: http://dlmf.nist.gov/8.19.E20 Encodings: TeX, pMML, png See also: Annotations for §8.19(ix), §8.19 and Ch.8
 8.19.21 $\frac{1}{x+n} ⓘ Symbols: $\mathrm{e}$: base of natural logarithm, $E_{\NVar{p}}\left(\NVar{z}\right)$: generalized exponential integral, $x$: real variable and $n$: nonnegative integer A&S Ref: 5.1.19 Permalink: http://dlmf.nist.gov/8.19.E21 Encodings: TeX, pMML, png See also: Annotations for §8.19(ix), §8.19 and Ch.8
 8.19.22 $\frac{\mathrm{d}}{\mathrm{d}x}\frac{E_{n}\left(x\right)}{E_{n-1}\left(x\right)% }>0.$

## §8.19(x) Integrals

 8.19.23 $\int_{z}^{\infty}E_{p-1}\left(t\right)\mathrm{d}t=E_{p}\left(z\right),$ $|\operatorname{ph}z|<\pi$,
 8.19.24 $\int_{0}^{\infty}e^{-at}E_{n}\left(t\right)\mathrm{d}t=\frac{(-1)^{n-1}}{a^{n}% }\left(\ln\left(1+a\right)+\sum_{k=1}^{n-1}\frac{(-1)^{k}a^{k}}{k}\right),$ $n=1,2,\dots$, $\Re a>-1$,
 8.19.25 $\int_{0}^{\infty}e^{-at}t^{b-1}E_{p}\left(t\right)\mathrm{d}t=\frac{\Gamma% \left(b\right)(1+a)^{-b}}{p+b-1}\*F\left(1,b;p+b;a/(1+a)\right),$ $\Re a>-1$, $\Re(p+b)>1$.
 8.19.26 $\int_{0}^{\infty}E_{p}\left(t\right)E_{q}\left(t\right)\mathrm{d}t=\frac{L(p)+% L(q)}{p+q-1},$ $p>0$, $q>0$, $p+q>1$,

where

 8.19.27 $L(p)=\int_{0}^{\infty}e^{-t}E_{p}\left(t\right)\mathrm{d}t=\frac{1}{2p}F\left(% 1,1;1+p;\tfrac{1}{2}\right),$ $p>0$.

For the hypergeometric function $F\left(a,b;c;z\right)$ see §15.2(i). When $p=1,2,3,\dots$, $L(p)$ can also be evaluated via (8.19.24).

For collections of integrals involving $E_{p}\left(z\right)$, especially for integer $p$, see Apelblat (1983, §§7.1–7.2) and LeCaine (1945).

## §8.19(xi) Further Generalizations

For higher-order generalized exponential integrals see Meijer and Baken (1987) and Milgram (1985).