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

Permalink:: http://dlmf.nist.gov/8.20

§8.20(i) Large $z$
§8.20(ii) Large $p$

§8.20(i) Large $z$

Notes:: See Olver (1991a).
Keywords:: generalized exponential integral
Permalink:: http://dlmf.nist.gov/8.20.i

8.20.1 $\mathop{E_{{p}}\/}\nolimits\!\left(z\right)=\frac{e^{{-z}}}{z}\left(\sum _{{k=0}}^{{n-1}}(-1)^{k}\frac{\left(p\right)_{{k}}}{z^{k}}+(-1)^{n}\frac{\left(p\right)_{{n}}e^{z}}{z^{{n-1}}}\mathop{E_{{n+p}}\/}\nolimits\!\left(z\right)\right),$ $n=1,2,3,\dots$ .

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $e$ : base of exponential function, $\mathop{E_{{p}}\/}\nolimits\!\left(z\right)$ : 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

As $z\to\infty$

8.20.2 $\mathop{E_{{p}}\/}\nolimits\!\left(z\right)\sim\frac{e^{{-z}}}{z}\sum _{{k=0}}^{\infty}(-1)^{k}\frac{\left(p\right)_{{k}}}{z^{k}},$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\frac{3}{2}\pi-\delta$ ,

Symbols:: $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $e$ : base of exponential function, $\mathop{E_{{p}}\/}\nolimits\!\left(z\right)$ : generalized exponential integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $z$ : 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

and

8.20.3 $\mathop{E_{{p}}\/}\nolimits\!\left(z\right)\sim\pm\frac{2\pi i}{\mathop{\Gamma\/}\nolimits\!\left(p\right)}e^{{\mp p\pi i}}z^{{p-1}}+\frac{e^{{-z}}}{z}\sum _{{k=0}}^{\infty}\frac{(-1)^{k}\left(p\right)_{{k}}}{z^{k}},$ $\tfrac{1}{2}\pi+\delta\leq\pm\mathop{\mathrm{ph}\/}\nolimits z\leq\tfrac{7}{2}\pi-\delta$ ,

Symbols:: $\mathop{\Gamma\/}\nolimits\!\left(z\right)$ : gamma function, $\left(a\right)_{{n}}$ : Pochhammer’s symbol, $e$ : base of exponential function, $\mathop{E_{{p}}\/}\nolimits\!\left(z\right)$ : generalized exponential integral, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality, $z$ : 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

$\delta$ 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 $\mathop{E_{{p}}\/}\nolimits\!\left(z\right)$ see §2.11(iii).

§8.20(ii) Large $p$

Notes:: See Gautschi (1959a).
A&S Ref:: 5.1.52 (This expansion is presented in a more general form.)
Keywords:: generalized exponential integral
Permalink:: http://dlmf.nist.gov/8.20.ii

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

8.20.4 $A_{{k+1}}(\lambda)=(1-2k\lambda)A_{k}(\lambda)+\lambda(\lambda+1)\frac{dA_{k}(\lambda)}{d\lambda},$ $k=0,1,2,\dots$ ,

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

so that $A_{k}(\lambda)$ is a polynomial in $\lambda$ of degree $k-1$ when $k\geq 1$ . 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
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Then as $p\to\infty$

8.20.6 $\mathop{E_{{p}}\/}\nolimits\!\left(\lambda p\right)\sim\frac{e^{{-\lambda p}}}{(\lambda+1)p}\sum _{{k=0}}^{\infty}\frac{A_{k}(\lambda)}{(\lambda+1)^{{2k}}}\frac{1}{p^{k}},$

Symbols:: $e$ : base of exponential function, $\mathop{E_{{p}}\/}\nolimits\!\left(z\right)$ : generalized exponential integral, $\sim$ : asymptotic equality, $p$ : parameter, $k$ : nonnegative integer and $A_{k}(\lambda)$
Permalink:: http://dlmf.nist.gov/8.20.E6
Encodings:: TeX, pMML, png

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).

§8.20 Asymptotic Expansions of

Contents

§8.20(i) Large

§8.20(ii) Large

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

§8.20(i) Large $z$

§8.20(ii) Large $p$