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

§6.13 Zeros

ⓘ

Keywords:: asymptotic expansion, cosine integrals, exponential integrals, sine integrals, zeros
Notes:: See Cody and Thacher (1969) for $x_{0}$ in (6.13.1).
Permalink:: http://dlmf.nist.gov/6.13
See also:: Annotations for Ch.6

The function $\operatorname{Ei}\left(x\right)$ 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 and Ch.6

$\operatorname{Ci}\left(x\right)$ and $\operatorname{si}\left(x\right)$ 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: $\sim$ : Poincaré asymptotic expansion, $c_{k}$ : zeros of $\operatorname{Ci}\left(x\right)$ and $s_{k}$ : zeros of $\operatorname{Si}\left(x\right)$ Referenced by: §6.13, §6.18(iii) Permalink: http://dlmf.nist.gov/6.13.E2 Encodings: TeX, pMML, png See also: Annotations for §6.13 and Ch.6

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).

6.12 Asymptotic Expansions 6.14 Integrals

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