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

§6.12 Asymptotic Expansions

Referenced by:: §6.18(i)
Permalink:: http://dlmf.nist.gov/6.12

§6.12(i) Exponential and Logarithmic Integrals
§6.12(ii) Sine and Cosine Integrals

§6.12(i) Exponential and Logarithmic Integrals

Notes:: For (6.12.2) see Olver (1997b, p. 227).
Keywords:: exponential integrals, logarithmic integral, sine integrals
Referenced by:: §2.11(vi), §6.12(ii)
Permalink:: http://dlmf.nist.gov/6.12.i

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

Symbols:: : base of exponential function, $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ : exponential integral, : : factorial, $\mathop{\mathrm{ph}\/}\nolimits$ : phase, $\sim$ : asymptotic equality and : complex variable
A&S Ref:: 5.1.51 (special case of)
Permalink:: http://dlmf.nist.gov/6.12.E1
Encodings:: TeX, pMML, png

When $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\frac{1}{2}\pi$ the remainder is bounded in magnitude by the first neglected term, and has the same sign when $\mathop{\mathrm{ph}\/}\nolimits z=0$ . When $\frac{1}{2}\pi\leq|\mathop{\mathrm{ph}\/}\nolimits z|<\pi$ the remainder term is bounded in magnitude by $\mathop{\csc\/}\nolimits\!\left(|\mathop{\mathrm{ph}\/}\nolimits z|\right)$ 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 .

6.12.2 $\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)\sim\frac{e^{{x}}}{x}\left(1+% \frac{1!}{x}+\frac{2!}{x^{2}}+\frac{3!}{x^{3}}+\cdots\right),$ $x\to+\infty$ .

Symbols:: : base of exponential function, $\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)$ : exponential integral, : : factorial, $\sim$ : asymptotic equality and : real variable
Referenced by:: §6.12(i), §6.12(i)
Permalink:: http://dlmf.nist.gov/6.12.E2
Encodings:: TeX, pMML, png

If the expansion is terminated at the 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 $\mathop{\mathrm{li}\/}\nolimits\!\left(x\right)$ as $x\to\infty$ is obtainable from (6.2.8) and (6.12.2).

§6.12(ii) Sine and Cosine 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 $-2\mathop{\mathrm{ph}\/}\nolimits z$ , and then replacing $|\tau+1|$ by its minimum value on the path.
Keywords:: auxiliary functions for sine and cosine integrals, cosine integrals, sine integrals
Permalink:: http://dlmf.nist.gov/6.12.ii

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

6.12.3 $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)\sim\frac{1}{z}\left(1-\frac{2!}% {z^{2}}+\frac{4!}{z^{4}}-\frac{6!}{z^{6}}+\cdots\right),$

Symbols:: : : factorial, $\sim$ : asymptotic equality, $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)$ : auxiliary function for sine and cosine integrals and : 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

6.12.4 $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)\sim\frac{1}{z^{2}}\left(1-\frac% {3!}{z^{2}}+\frac{5!}{z^{4}}-\frac{7!}{z^{6}}+\cdots\right),$

Symbols:: : : factorial, $\sim$ : asymptotic equality, $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)$ : auxiliary function for sine and cosine integrals and : 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

as $z\to\infty$ in $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi-\delta\thinspace(<\pi)$ .

The remainder terms are given by

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

Symbols:: : : factorial, $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)$ : auxiliary function for sine and cosine integrals, : complex variable, : nonnegative integer and $R_{n}^{{(\mathop{\mathrm{f}\/}\nolimits)}}(z)$ : remainder term
Referenced by:: §6.12(ii)
Permalink:: http://dlmf.nist.gov/6.12.E5
Encodings:: TeX, pMML, png

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

Symbols:: : : factorial, $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)$ : auxiliary function for sine and cosine integrals, : complex variable, : nonnegative integer and $R_{n}^{{(\mathop{\mathrm{g}\/}\nolimits)}}(z)$ : remainder term
Permalink:: http://dlmf.nist.gov/6.12.E6
Encodings:: TeX, pMML, png

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

6.12.7 $R_{n}^{{(\mathop{\mathrm{f}\/}\nolimits)}}(z)=(-1)^{n}\int_{0}^{\infty}\frac{e% ^{{-zt}}t^{{2n}}}{t^{2}+1}dt,$

Symbols:: : differential of , : base of exponential function, $\int$ : integral, $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)$ : auxiliary function for sine and cosine integrals, : complex variable, : nonnegative integer and $R_{n}^{{(\mathop{\mathrm{f}\/}\nolimits)}}(z)$ : remainder term
Referenced by:: §6.12(ii)
Permalink:: http://dlmf.nist.gov/6.12.E7
Encodings:: TeX, pMML, png

6.12.8 $R_{n}^{{(\mathop{\mathrm{g}\/}\nolimits)}}(z)=(-1)^{n}\int_{0}^{\infty}\frac{e% ^{{-zt}}t^{{2n+1}}}{t^{2}+1}dt.$

Symbols:: : differential of , : base of exponential function, $\int$ : integral, $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)$ : auxiliary function for sine and cosine integrals, : complex variable, : nonnegative integer and $R_{n}^{{(\mathop{\mathrm{g}\/}\nolimits)}}(z)$ : remainder term
Referenced by:: §6.12(ii)
Permalink:: http://dlmf.nist.gov/6.12.E8
Encodings:: TeX, pMML, png

When $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\tfrac{1}{4}\pi$ , 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 $\mathop{\mathrm{ph}\/}\nolimits z=0$ . When $\frac{1}{4}\pi\leq|\mathop{\mathrm{ph}\/}\nolimits z|<\frac{1}{2}\pi$ the remainders are bounded in magnitude by $\mathop{\csc\/}\nolimits\!\left(2|\mathop{\mathrm{ph}\/}\nolimits z|\right)$ 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).

© 2010–2013 NIST / Privacy Policy / Disclaimer / Feedback; Version 1.0.6; Release date 2013-05-06 Math-Image version (See MathML) . A Printed Companionis available. 6.11 Relations to Other Functions 6.13 Zeros

§6.12 Asymptotic Expansions

Contents

§6.12(i) Exponential and Logarithmic Integrals

§6.12(ii) Sine and Cosine Integrals