6 Exponential, Logarithmic, Sine, and Cosine Integrals Properties 6.11 Relations to Other Functions 6.13 Zeros

§6.12 Asymptotic Expansions

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Referenced by:: §6.18(i)
Permalink:: http://dlmf.nist.gov/6.12
See also:: Annotations for Ch.6

§6.12(i) Exponential and Logarithmic Integrals

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Keywords:: asymptotic expansion, asymptotic expansions, exponential integrals, exponentially-improved, logarithmic integral, re-expansion of remainder term, sine integrals
Notes:: For (6.12.2) see Olver (1997b, p. 227).
Referenced by:: §2.11(vi), §6.12(ii)
Permalink:: http://dlmf.nist.gov/6.12.i
See also:: Annotations for §6.12 and Ch.6

6.12.1		$E_{1}\left(z\right)\sim\frac{e^{-z}}{z}\left(1-\frac{1!}{z}+\frac{2!}{z^{2}}-% \frac{3!}{z^{3}}+\cdots\right),$
$z\to\infty$ , $\|\operatorname{ph}z\|\leq\tfrac{3}{2}\pi-\delta(<\tfrac{3}{2}\pi)$ .
ⓘ Symbols: $\sim$ : Poincaré asymptotic expansion, $\pi$ : the ratio of the circumference of a circle to its diameter, $\mathrm{e}$ : base of natural logarithm, $E_{1}\left(\NVar{z}\right)$ : exponential integral, $!$ : factorial (as in $n!$ ), $\operatorname{ph}$ : 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 $|\operatorname{ph}z|\leq\frac{1}{2}\pi$ 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		$\operatorname{Ei}\left(x\right)\sim\frac{e^{x}}{x}\left(1+\frac{1!}{x}+\frac{2% !}{x^{2}}+\frac{3!}{x^{3}}+\cdots\right),$
$x\to+\infty$ .
ⓘ Symbols: $\sim$ : Poincaré asymptotic expansion, $\mathrm{e}$ : base of natural logarithm, $\operatorname{Ei}\left(\NVar{x}\right)$ : 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 $\operatorname{li}\left(x\right)$ as $x\to\infty$ is obtainable from (6.2.8) and (6.12.2).

§6.12(ii) Sine and Cosine Integrals

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Keywords:: asymptotic expansions, auxiliary functions for sine and cosine integrals, cosine integrals, exponentially-improved, sine integrals
Notes:: (6.12.3) and (6.12.4) follow from (6.7.13) and (6.7.14) by applying Watson’s lemma (§2.4(i)). (6.12.5)–(6.12.8) follow from (6.7.13), (6.7.14), and the identity $(t^{2}+1)^{-1}=\sum_{m=0}^{n-1}(-1)^{m}t^{2m}+(-1)^{n}t^{2n}(t^{2}+1)^{-1}$ . The error bounds are obtained by setting $t=\sqrt{\tau}$ in (6.12.7) and (6.12.8), rotating the integration path in the $\tau$ -plane through an angle $-2\operatorname{ph}z$ , and then replacing $|\tau+1|$ by its minimum value on the path.
Permalink:: http://dlmf.nist.gov/6.12.ii
See also:: Annotations for §6.12 and Ch.6

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	$\displaystyle\mathrm{f}\left(z\right)$	$\displaystyle=\frac{1}{z}\sum_{m=0}^{n-1}(-1)^{m}\frac{(2m)!}{z^{2m}}+R_{n}^{(% \mathrm{f})}(z),$
ⓘ Symbols: $!$ : factorial (as in $n!$ ), $\mathrm{f}\left(\NVar{z}\right)$ : 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	$\displaystyle\mathrm{g}\left(z\right)$	$\displaystyle=\frac{1}{z^{2}}\sum_{m=0}^{n-1}(-1)^{m}\frac{(2m+1)!}{z^{2m}}+R_% {n}^{(\mathrm{g})}(z),$
ⓘ Symbols: $!$ : factorial (as in $n!$ ), $\mathrm{g}\left(\NVar{z}\right)$ : 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 $n=0,1,2,\dots$ ,

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).

6.11 Relations to Other Functions 6.13 Zeros

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§6.12 Asymptotic Expansions

Contents

§6.12(i) Exponential and Logarithmic Integrals

§6.12(ii) Sine and Cosine Integrals

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