§6.12 Asymptotic Expansions

§6.12(i) Exponential and Logarithmic Integrals

 6.12.1 $E_{1}\left(z\right)\sim\frac{e^{-z}}{z}\left(1-\frac{1!}{z}+\frac{2!}{z^{2}}-% \frac{3!}{z^{3}}+\cdots\right),$ $z\to\infty$, $|\operatorname{ph}z|\leq\tfrac{3}{2}\pi-\delta(<\tfrac{3}{2}\pi)$.

When $|\operatorname{ph}z|\leq\frac{1}{2}\pi$ the remainder is bounded in magnitude by the first neglected term, and has the same sign when $\operatorname{ph}z=0$. When $\frac{1}{2}\pi\leq|\operatorname{ph}z|<\pi$ the remainder term is bounded in magnitude by $\csc\left(|\operatorname{ph}z|\right)$ 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 $\operatorname{Ei}\left(x\right)\sim\frac{e^{x}}{x}\left(1+\frac{1!}{x}+\frac{2% !}{x^{2}}+\frac{3!}{x^{3}}+\cdots\right),$ $x\to+\infty$.

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 $\operatorname{li}\left(x\right)$ as $x\to\infty$ is obtainable from (6.2.8) and (6.12.2).

§6.12(ii) Sine and Cosine Integrals

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

 6.12.3 $\displaystyle\mathrm{f}\left(z\right)$ $\displaystyle\sim\frac{1}{z}\left(1-\frac{2!}{z^{2}}+\frac{4!}{z^{4}}-\frac{6!% }{z^{6}}+\cdots\right),$ ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $\mathrm{f}\left(\NVar{z}\right)$: 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 $\displaystyle\mathrm{g}\left(z\right)$ $\displaystyle\sim\frac{1}{z^{2}}\left(1-\frac{3!}{z^{2}}+\frac{5!}{z^{4}}-% \frac{7!}{z^{6}}+\cdots\right),$ ⓘ Symbols: $\sim$: Poincaré asymptotic expansion, $!$: factorial (as in $n!$), $\mathrm{g}\left(\NVar{z}\right)$: 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 $z\to\infty$ in $|\operatorname{ph}z|\leq\pi-\delta\thinspace(<\pi)$.

The remainder terms are given by

 6.12.5 $\displaystyle\mathrm{f}\left(z\right)$ $\displaystyle=\frac{1}{z}\sum_{m=0}^{n-1}(-1)^{m}\frac{(2m)!}{z^{2m}}+R_{n}^{(% \mathrm{f})}(z),$ 6.12.6 $\displaystyle\mathrm{g}\left(z\right)$ $\displaystyle=\frac{1}{z^{2}}\sum_{m=0}^{n-1}(-1)^{m}\frac{(2m+1)!}{z^{2m}}+R_% {n}^{(\mathrm{g})}(z),$

where, for $n=0,1,2,\dots$,

 6.12.7 $\displaystyle R_{n}^{(\mathrm{f})}(z)$ $\displaystyle=(-1)^{n}\int_{0}^{\infty}\frac{e^{-zt}t^{2n}}{t^{2}+1}\,\mathrm{% d}t,$ ⓘ Defines: $R_{n}^{(\mathrm{f})}(z)$: remainder term (locally) Symbols: $\,\mathrm{d}\NVar{x}$: 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 $\displaystyle R_{n}^{(\mathrm{g})}(z)$ $\displaystyle=(-1)^{n}\int_{0}^{\infty}\frac{e^{-zt}t^{2n+1}}{t^{2}+1}\,% \mathrm{d}t.$ ⓘ Defines: $R_{n}^{(\mathrm{g})}(z)$: remainder term (locally) Symbols: $\,\mathrm{d}\NVar{x}$: 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 $|\operatorname{ph}z|\leq\tfrac{1}{4}\pi$, 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 $\operatorname{ph}z=0$. When $\frac{1}{4}\pi\leq|\operatorname{ph}z|<\frac{1}{2}\pi$ the remainders are bounded in magnitude by $\csc\left(2|\operatorname{ph}z|\right)$ 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).