8 Incomplete Gamma and Related Functions Related Functions 8.19 Generalized Exponential Integral 8.21 Generalized Sine and Cosine Integrals

§8.20 Asymptotic Expansions of $E_{p}\left(z\right)$

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Permalink:: http://dlmf.nist.gov/8.20
See also:: Annotations for Ch.8

§8.20(i) Large $z$

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Keywords:: asymptotic expansions, exponentially-improved, generalized exponential integral, large variable
Notes:: See Olver (1991a).
Permalink:: http://dlmf.nist.gov/8.20.i
See also:: Annotations for §8.20 and Ch.8

8.20.1		$E_{p}\left(z\right)=\frac{e^{-z}}{z}\left(\sum_{k=0}^{n-1}(-1)^{k}\frac{{\left% (p\right)_{k}}}{z^{k}}+(-1)^{n}\frac{{\left(p\right)_{n}}e^{z}}{z^{n-1}}E_{n+p% }\left(z\right)\right),$
$n=1,2,3,\dots$ .
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : Pochhammer’s symbol (or shifted factorial), $\mathrm{e}$ : base of natural logarithm, $E_{\NVar{p}}\left(\NVar{z}\right)$ : 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 $z\to\infty$

8.20.2		$E_{p}\left(z\right)\sim\frac{e^{-z}}{z}\sum_{k=0}^{\infty}(-1)^{k}\frac{{\left% (p\right)_{k}}}{z^{k}},$
$\|\operatorname{ph}z\|\leq\frac{3}{2}\pi-\delta$ ,
ⓘ Symbols: ${\left(\NVar{a}\right)_{\NVar{n}}}$ : 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, $E_{\NVar{p}}\left(\NVar{z}\right)$ : generalized exponential integral, $\operatorname{ph}$ : 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		$E_{p}\left(z\right)\sim\pm\frac{2\pi i}{\Gamma\left(p\right)}e^{\mp p\pi i}z^{% p-1}+\frac{e^{-z}}{z}\sum_{k=0}^{\infty}\frac{(-1)^{k}{\left(p\right)_{k}}}{z^% {k}},$
$\tfrac{1}{2}\pi+\delta\leq\pm\operatorname{ph}z\leq\tfrac{7}{2}\pi-\delta$ ,
ⓘ Symbols: $\Gamma\left(\NVar{z}\right)$ : gamma function, ${\left(\NVar{a}\right)_{\NVar{n}}}$ : 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, $E_{\NVar{p}}\left(\NVar{z}\right)$ : generalized exponential integral, $\mathrm{i}$ : imaginary unit, $\operatorname{ph}$ : 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).

§8.20(ii) Large $p$

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Keywords:: asymptotic expansions, generalized exponential integral, large parameter
A&S Ref:: 5.1.52 (This expansion is presented in a more general form.)
Notes:: See Gautschi (1959a).
Permalink:: http://dlmf.nist.gov/8.20.ii
See also:: Annotations for §8.20 and Ch.8

For $x\geq 0$ and $p>1$ let $x=\lambda p$ and define $A_{0}(\lambda)=1$ ,

8.20.4		$A_{k+1}(\lambda)=(1-2k\lambda)A_{k}(\lambda)+\lambda(\lambda+1)\frac{\mathrm{d% }A_{k}(\lambda)}{\mathrm{d}\lambda},$
$k=0,1,2,\dots$ ,
ⓘ Symbols: $\frac{\mathrm{d}\NVar{f}}{\mathrm{d}\NVar{x}}$ : 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 $k\geq 1$ . In particular,

8.20.5	$\displaystyle A_{1}(\lambda)$	$\displaystyle=1,$
	$\displaystyle A_{2}(\lambda)$	$\displaystyle=1-2\lambda,$
	$\displaystyle A_{3}(\lambda)$	$\displaystyle=1-8\lambda+6\lambda^{2}.$
ⓘ 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 $p\to\infty$

8.20.6		$E_{p}\left(\lambda p\right)\sim\frac{e^{-\lambda p}}{(\lambda+1)p}\sum_{k=0}^{% \infty}\frac{A_{k}(\lambda)}{(\lambda+1)^{2k}}\frac{1}{p^{k}},$
ⓘ Symbols: $\sim$ : Poincaré asymptotic expansion, $\mathrm{e}$ : base of natural logarithm, $E_{\NVar{p}}\left(\NVar{z}\right)$ : 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 $\lambda\in[0,\infty)$ .

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

8.19 Generalized Exponential Integral 8.21 Generalized Sine and Cosine Integrals

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§8.20 Asymptotic Expansions of Ep⁡(z)

Contents

§8.20(i) Large z

§8.20(ii) Large p

§8.20 Asymptotic Expansions of $E_{p}\left(z\right)$

§8.20(i) Large $z$

§8.20(ii) Large $p$