6 Exponential, Logarithmic, Sine, and Cosine Integrals Properties 6.12 Asymptotic Expansions 6.14 Integrals

§6.13 Zeros

Notes:: See Cody and Thacher (1969) for $x_{0}$ in (6.13.1).
Keywords:: cosine integrals, exponential integrals, sine integrals
Permalink:: http://dlmf.nist.gov/6.13

The function $\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)$ has one real zero $x_{0}$ , given by

6.13.1 $x_{0}=0.37250\;74107\;81366\;63446\;19918\;66580\dots.$

Notes:: For more digits see OEIS Sequence A091723; see also Sloane (2003).
Symbols:: : real variable
Referenced by:: §6.13
Permalink:: http://dlmf.nist.gov/6.13.E1
Encodings:: TeX, pMML, png

$\mathop{\mathrm{Ci}\/}\nolimits\!\left(x\right)$ and $\mathop{\mathrm{si}\/}\nolimits\!\left(x\right)$ each have an infinite number of positive real zeros, which are denoted by $c_{k}$ , $s_{k}$ , respectively, arranged in ascending order of absolute value for $k=0,1,2,\dots$ . Values of $c_{1}$ and $c_{2}$ to 30D are given by MacLeod (1996b).

As $k\to\infty$ ,

6.13.2 $c_{k},s_{k}\sim\alpha+\frac{1}{\alpha}-\frac{16}{3}\frac{1}{\alpha^{3}}+\frac{% 1673}{15}\frac{1}{\alpha^{5}}-\frac{5\;07746}{105}\frac{1}{\alpha^{7}}+\cdots,$

Symbols:: $\sim$ : asymptotic equality, $c_{k}$ : zeros of $\mathop{\mathrm{Ci}\/}\nolimits\!\left(x\right)$ and $s_{k}$ : zeros of $\mathop{\mathrm{Si}\/}\nolimits\!\left(x\right)$
Referenced by:: §6.13, §6.18(iii)
Permalink:: http://dlmf.nist.gov/6.13.E2
Encodings:: TeX, pMML, png

where $\alpha=k\pi$ for $c_{k}$ , and $\alpha=(k+\frac{1}{2})\pi$ for $s_{k}$ . For these results, together with the next three terms in (6.13.2), see MacLeod (2002a). See also Riekstynš (1991, pp. 176–177).

© 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. 6.12 Asymptotic Expansions 6.14 Integrals