§6.13 Zeros

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

 6.13.1 $x_{0}=0.37250\;74107\;81366\;63446\;19918\;66580\dots.$ Symbols: $x$: real variable Notes: For more digits see OEIS Sequence A091723; see also Sloane (2003). Referenced by: §6.13 Permalink: http://dlmf.nist.gov/6.13.E1 Encodings: TeX, pMML, png See also: Annotations for 6.13

$\mathop{\mathrm{Ci}\/}\nolimits\!\left(x\right)$ 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 $c_{k},s_{k}\sim\alpha+\frac{1}{\alpha}-\frac{16}{3}\frac{1}{\alpha^{3}}+\frac{% 1673}{15}\frac{1}{\alpha^{5}}-\frac{5\;07746}{105}\frac{1}{\alpha^{7}}+\cdots,$

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).