§6.13 Zeros

The function $\mathrm{Ei}\left(x\right)$ has one real zero $x_0$, given by

$\mathrm{Ci}\left(x\right)$ and $\mathrm{si}\left(x\right)$ 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 $k=0,1,2,\mathrm{\dots }$. Values of $c_1$ and $c_2$ to 30D are given by MacLeod (1996b).

As $k\to \mathrm{\infty }$,

where $\alpha =k\pi$ for $c_k$, and $\alpha =(k+\frac{1}{2})\pi$ for $s_k$. In (6.13.2), the remainder after $n$th terms does not exceed the $(n+1)$th term in absolute value and has the same sign. For these results, together with the next three terms in (6.13.2), see MacLeod (2002a). See also Riekstynš (1991, pp. 176–177), Nemes (2025).