§8.20 Asymptotic Expansions of $E_p\left(z\right)$

8.20.1		$$E_p\left(z\right)=\frac{{\mathrm{e}}^{-z}}{z}\left(\sum _{k=0}^{n-1}{(-1)}^k\frac{{\left(p\right)}_k}{z^k}+{(-1)}^n\frac{{\left(p\right)}_n{\mathrm{e}}^z}{z^{n-1}}{E}_{n+p}\left(z\right)\right),$$
$n=1,2,3,\mathrm{\dots }$.
ⓘ Symbols: ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $z$: complex variable, $p$: parameter, $k$: nonnegative integer and $n$: nonnegative integer A&S Ref: 5.1.51 Permalink: http://dlmf.nist.gov/8.20.E1 Encodings: TeX, pMML, png See also: Annotations for §8.20(i), §8.20 and Ch.8

As $z\to \mathrm{\infty }$

8.20.2		$$E_p\left(z\right)\sim \frac{{\mathrm{e}}^{-z}}{z}\sum _{k=0}^{\mathrm{\infty }}{(-1)}^k\frac{{\left(p\right)}_k}{z^k},$$
$|\mathrm{ph}z|\le \frac{3}{2}\pi -\delta$,
ⓘ Symbols: ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\mathrm{ph}$: phase, $z$: complex variable, $p$: parameter, $k$: nonnegative integer and $\delta$: small positive constant Referenced by: §8.20(i) Permalink: http://dlmf.nist.gov/8.20.E2 Encodings: TeX, pMML, png See also: Annotations for §8.20(i), §8.20 and Ch.8

and

8.20.3		$$E_p\left(z\right)\sim \pm \frac{2\pi \mathrm{i}}{\mathrm{\Gamma }\left(p\right)}{\mathrm{e}}^{\mp p\pi \mathrm{i}}{z}^{p-1}+\frac{{\mathrm{e}}^{-z}}{z}\sum _{k=0}^{\mathrm{\infty }}\frac{{(-1)}^k{\left(p\right)}_k}{z^k},$$
$\frac{1}{2}\pi +\delta \le \pm \mathrm{ph}z\le \frac{7}{2}\pi -\delta$,
ⓘ Symbols: $\mathrm{\Gamma }\left(z\right)$: gamma function, ${\left(a\right)}_n$: Pochhammer’s symbol (or shifted factorial), $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $\mathrm{i}$: imaginary unit, $\mathrm{ph}$: phase, $z$: complex variable, $p$: parameter, $k$: nonnegative integer and $\delta$: small positive constant Referenced by: §2.11(ii), §8.20(i) Permalink: http://dlmf.nist.gov/8.20.E3 Encodings: TeX, pMML, png See also: Annotations for §8.20(i), §8.20 and Ch.8

$\delta$ again denoting an arbitrary small positive constant. 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).

For an exponentially-improved asymptotic expansion of $E_p\left(z\right)$ see §2.11(iii).

For $x\ge 0$ and $p>1$ let $x=\lambda p$ and define $A_0(\lambda )=1$,

8.20.4		$$A_{k+1}(\lambda )=(1-2k\lambda )A_k(\lambda )+\lambda (\lambda +1)\frac{d{A}_k(\lambda )}{d\lambda },$$
$k=0,1,2,\mathrm{\dots }$,
ⓘ Symbols: $\frac{df}{dx}$: derivative of $f$ with respect to $x$, $k$: nonnegative integer and $A_k(\lambda )$ Permalink: http://dlmf.nist.gov/8.20.E4 Encodings: TeX, pMML, png See also: Annotations for §8.20(ii), §8.20 and Ch.8

so that $A_k(\lambda )$ is a polynomial in $\lambda$ of degree $k-1$ when $k\ge 1$. In particular,

8.20.5		$A_1(\lambda )$	$=1,$
		$A_2(\lambda )$	$=1-2\lambda ,$
		$A_3(\lambda )$	$=1-8\lambda +6{\lambda }^2.$
ⓘ Symbols: $A_k(\lambda )$ Permalink: http://dlmf.nist.gov/8.20.E5 Encodings: TeX, TeX, TeX, pMML, pMML, pMML, png, png, png See also: Annotations for §8.20(ii), §8.20 and Ch.8

Then as $p\to \mathrm{\infty }$

8.20.6		$$E_p\left(\lambda p\right)\sim \frac{{\mathrm{e}}^{-\lambda p}}{(\lambda +1)p}\sum _{k=0}^{\mathrm{\infty }}\frac{A_k(\lambda )}{{(\lambda +1)}^{2k}}\frac{1}{p^k},$$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $E_p\left(z\right)$: generalized exponential integral, $p$: parameter, $k$: nonnegative integer and $A_k(\lambda )$ Permalink: http://dlmf.nist.gov/8.20.E6 Encodings: TeX, pMML, png See also: Annotations for §8.20(ii), §8.20 and Ch.8

uniformly for $\lambda \in [0,\mathrm{\infty })$.

For further information, including extensions to complex values of $x$ and $p$, see Temme (1994b, §4) and Dunster (1996b, 1997).