§6.12 Asymptotic Expansions

6.12.1		$$E_1\left(z\right)\sim \frac{{\mathrm{e}}^{-z}}{z}\left(1-\frac{1!}{z}+\frac{2!}{z^2}-\frac{3!}{z^3}+\mathrm{\cdots }\right),$$
$z\to \mathrm{\infty }$, .
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $E_1\left(z\right)$: exponential integral, $!$: factorial (as in $n!$), $\mathrm{ph}$: phase and $z$: complex variable A&S Ref: 5.1.51 (special case of) Permalink: http://dlmf.nist.gov/6.12.E1 Encodings: TeX, pMML, png See also: Annotations for §6.12(i), §6.12 and Ch.6

When $|\mathrm{ph}z|\le \frac{1}{2}\pi$ the remainder is bounded in magnitude by the first neglected term, and has the same sign when $\mathrm{ph}z=0$. When the remainder term is bounded in magnitude by $\csc \left(|\mathrm{ph}z|\right)$ times the first neglected term. For these and other error bounds see Olver (1997b, pp. 109–112) with $\alpha =0$.

For re-expansions of the remainder term leading to larger sectors of validity, exponential improvement, and a smooth interpretation of the Stokes phenomenon, see §§2.11(ii)–2.11(iv), with $p=1$.

6.12.2		$$\mathrm{Ei}\left(x\right)\sim \frac{{\mathrm{e}}^x}{x}\left(1+\frac{1!}{x}+\frac{2!}{x^2}+\frac{3!}{x^3}+\mathrm{\cdots }\right),$$
$x\to +\mathrm{\infty }$.
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $\mathrm{e}$: base of natural logarithm, $\mathrm{Ei}\left(x\right)$: exponential integral, $!$: factorial (as in $n!$) and $x$: real variable Referenced by: §6.12(i), §6.12(i) Permalink: http://dlmf.nist.gov/6.12.E2 Encodings: TeX, pMML, png See also: Annotations for §6.12(i), §6.12 and Ch.6

If the expansion is terminated at the $n$th term, then the remainder term is bounded by $1+\chi (n+1)$ times the next term. For the function $\chi$ see §9.7(i).

The asymptotic expansion of $\mathrm{li}\left(x\right)$ as $x\to \mathrm{\infty }$ is obtainable from (6.2.8) and (6.12.2).

The asymptotic expansions of $\mathrm{Si}\left(z\right)$ and $\mathrm{Ci}\left(z\right)$ are given by (6.2.19), (6.2.20), together with

6.12.3		$\mathrm{f}\left(z\right)$	$\sim {\displaystyle \frac{1}{z}}\left(1-{\displaystyle \frac{2!}{z^2}}+{\displaystyle \frac{4!}{z^4}}-{\displaystyle \frac{6!}{z^6}}+\mathrm{\cdots }\right),$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable A&S Ref: 5.2.34 Referenced by: §6.12(ii), §6.12(ii) Permalink: http://dlmf.nist.gov/6.12.E3 Encodings: TeX, pMML, png See also: Annotations for §6.12(ii), §6.12 and Ch.6
6.12.4		$\mathrm{g}\left(z\right)$	$\sim {\displaystyle \frac{1}{z^2}}\left(1-{\displaystyle \frac{3!}{z^2}}+{\displaystyle \frac{5!}{z^4}}-{\displaystyle \frac{7!}{z^6}}+\mathrm{\cdots }\right),$
ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals and $z$: complex variable A&S Ref: 5.2.35 Referenced by: §6.12(ii), §6.12(ii) Permalink: http://dlmf.nist.gov/6.12.E4 Encodings: TeX, pMML, png See also: Annotations for §6.12(ii), §6.12 and Ch.6

as $z\to \mathrm{\infty }$ in .

The remainder terms are given by

6.12.5		$\mathrm{f}\left(z\right)$	$={\displaystyle \frac{1}{z}}{\displaystyle \sum _{m=0}^{n-1}}{(-1)}^m{\displaystyle \frac{(2m)!}{z^{2m}}}+R_n^{(\mathrm{f})}(z),$
ⓘ Symbols: $!$: factorial (as in $n!$), $\mathrm{f}\left(z\right)$: auxiliary function for sine and cosine integrals, $z$: complex variable, $n$: nonnegative integer and $R_n^{(\mathrm{f})}(z)$: remainder term Referenced by: §6.12(ii) Permalink: http://dlmf.nist.gov/6.12.E5 Encodings: TeX, pMML, png See also: Annotations for §6.12(ii), §6.12 and Ch.6
6.12.6		$\mathrm{g}\left(z\right)$	$={\displaystyle \frac{1}{z^2}}{\displaystyle \sum _{m=0}^{n-1}}{(-1)}^m{\displaystyle \frac{(2m+1)!}{z^{2m}}}+R_n^{(\mathrm{g})}(z),$
ⓘ Symbols: $!$: factorial (as in $n!$), $\mathrm{g}\left(z\right)$: auxiliary function for sine and cosine integrals, $z$: complex variable, $n$: nonnegative integer and $R_n^{(\mathrm{g})}(z)$: remainder term Permalink: http://dlmf.nist.gov/6.12.E6 Encodings: TeX, pMML, png See also: Annotations for §6.12(ii), §6.12 and Ch.6

where, for $n=0,1,2,\mathrm{\dots }$,

6.12.7		$R_n^{(\mathrm{f})}(z)$	$={(-1)}^n{\displaystyle {\int }_0^{\mathrm{\infty }}}{\displaystyle \frac{{\mathrm{e}}^{-zt}{t}^{2n}}{t^2+1}}dt,$
ⓘ Defines: $R_n^{(\mathrm{f})}(z)$: remainder term (locally) Symbols: $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $z$: complex variable and $n$: nonnegative integer Referenced by: §6.12(ii) Permalink: http://dlmf.nist.gov/6.12.E7 Encodings: TeX, pMML, png See also: Annotations for §6.12(ii), §6.12 and Ch.6
6.12.8		$R_n^{(\mathrm{g})}(z)$	$={(-1)}^n{\displaystyle {\int }_0^{\mathrm{\infty }}}{\displaystyle \frac{{\mathrm{e}}^{-zt}{t}^{2n+1}}{t^2+1}}dt.$
ⓘ Defines: $R_n^{(\mathrm{g})}(z)$: remainder term (locally) Symbols: $dx$: differential of $x$, $\mathrm{e}$: base of natural logarithm, $\int$: integral, $z$: complex variable and $n$: nonnegative integer Referenced by: §6.12(ii) Permalink: http://dlmf.nist.gov/6.12.E8 Encodings: TeX, pMML, png See also: Annotations for §6.12(ii), §6.12 and Ch.6

When $|\mathrm{ph}z|\le \frac{1}{4}\pi$, these remainders are bounded in magnitude by the first neglected terms in (6.12.3) and (6.12.4), respectively, and have the same signs as these terms when $\mathrm{ph}z=0$. When the remainders are bounded in magnitude by $\csc \left(2|\mathrm{ph}z|\right)$ times the first neglected terms.

For other phase ranges use (6.4.6) and (6.4.7). For exponentially-improved asymptotic expansions, use (6.5.5), (6.5.6), and §6.12(i).