# §8.19 Generalized Exponential Integral

## §8.19(i) Definition and Integral Representations

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

 8.19.1 $\mathop{E_{p}\/}\nolimits\!\left(z\right)=z^{p-1}\mathop{\Gamma\/}\nolimits\!% \left(1-p,z\right).$ Defines: $\mathop{E_{\NVar{p}}\/}\nolimits\!\left(\NVar{z}\right)$: generalized exponential integral Symbols: $\mathop{\Gamma\/}\nolimits\!\left(\NVar{a},\NVar{z}\right)$: 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)

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

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

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

### Other Integral Representations

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

Integral representations of Mellin–Barnes type for $\mathop{E_{p}\/}\nolimits\!\left(z\right)$ 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 $\mathop{E_{0}\/}\nolimits\!\left(z\right)=z^{-1}e^{-z},$ $z\neq 0$, Symbols: $\mathrm{e}$: base of exponential function, $\mathop{E_{\NVar{p}}\/}\nolimits\!\left(\NVar{z}\right)$: 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)
 8.19.6 $\mathop{E_{p}\/}\nolimits\!\left(0\right)=\frac{1}{p-1},$ $\Re{p}>1$, Symbols: $\mathop{E_{\NVar{p}}\/}\nolimits\!\left(\NVar{z}\right)$: 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)
 8.19.7 $\mathop{E_{n}\/}\nolimits\!\left(z\right)=\frac{(-z)^{n-1}}{(n-1)!}\mathop{E_{% 1}\/}\nolimits\!\left(z\right)+\frac{e^{-z}}{(n-1)!}\sum_{k=0}^{n-2}(n-k-2)!(-% z)^{k},$ $n=2,3,\dots$.

## §8.19(iv) Series Expansions

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

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

and

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

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

When $p\in\mathbb{C}$

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

again with $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi$ 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 $p\mathop{E_{p+1}\/}\nolimits\!\left(z\right)+z\mathop{E_{p}\/}\nolimits\!\left% (z\right)=e^{-z}.$
 8.19.13 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}\mathop{E_{p}\/}\nolimits\!\left(z\right)$ $\displaystyle=-\mathop{E_{p-1}\/}\nolimits\!\left(z\right),$ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$: derivative of $f$ with respect to $x$, $\mathop{E_{\NVar{p}}\/}\nolimits\!\left(\NVar{z}\right)$: 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) 8.19.14 $\displaystyle\frac{\mathrm{d}}{\mathrm{d}z}(e^{z}\mathop{E_{p}\/}\nolimits\!% \left(z\right))$ $\displaystyle=e^{z}\mathop{E_{p}\/}\nolimits\!\left(z\right)\left(1+\frac{p-1}% {z}\right)-\frac{1}{z}.$

### $p$-Derivatives

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

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

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 $\mathop{E_{p}\/}\nolimits\!\left(z\right)=z^{p-1}e^{-z}\mathop{U\/}\nolimits\!% \left(p,p,z\right).$

For $\mathop{U\/}\nolimits\!\left(a,b,z\right)$ see §13.2(i).

## §8.19(vii) Continued Fraction

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

## §8.19(viii) Analytic Continuation

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

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

## §8.19(ix) Inequalities

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

 8.19.19 $\frac{n-1}{n}\mathop{E_{n}\/}\nolimits\!\left(x\right)<\mathop{E_{n+1}\/}% \nolimits\!\left(x\right)<\mathop{E_{n}\/}\nolimits\!\left(x\right),$ Symbols: $\mathop{E_{\NVar{p}}\/}\nolimits\!\left(\NVar{z}\right)$: 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)
 8.19.20 $\left(\mathop{E_{n}\/}\nolimits\!\left(x\right)\right)^{2}<\mathop{E_{n-1}\/}% \nolimits\!\left(x\right)\mathop{E_{n+1}\/}\nolimits\!\left(x\right),$ Symbols: $\mathop{E_{\NVar{p}}\/}\nolimits\!\left(\NVar{z}\right)$: 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)
 8.19.21 $\frac{1}{x+n} Symbols: $\mathrm{e}$: base of exponential function, $\mathop{E_{\NVar{p}}\/}\nolimits\!\left(\NVar{z}\right)$: 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)
 8.19.22 $\frac{\mathrm{d}}{\mathrm{d}x}\frac{\mathop{E_{n}\/}\nolimits\!\left(x\right)}% {\mathop{E_{n-1}\/}\nolimits\!\left(x\right)}>0.$

## §8.19(x) Integrals

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

where

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

For the hypergeometric function $\mathop{F\/}\nolimits\!\left(a,b;c;z\right)$ 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 $\mathop{E_{p}\/}\nolimits\!\left(z\right)$, 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).