Asymptotic expansions for the zeros of certain special functions (Q1612402)

In this paper asymptotic formulae for the zeros of the cosine-integral \(Ci(x)\), the Struve function \(H_0(x)\) as well as the Kelvin functions are derived showing an acceptable degree of accuracy. By using the standard technique by \textit{F. W. J. Olver} [Asymptotics and special functions, Chapter 1, Academic Press, New York (1974; Zbl 0303.41035)] the obtained expression for \(Ci(x)\) gives more than 10D accuracy for all roots beyond the ninth. The corresponding expansion to \(H_0(x)\) [see \textit{M. Abramowitz} and \textit{I. A. Stegun}, Handbook of mathematical functions (Reprint of the 1972 ed.), Chapter 12, J. Wiley Publ., New York (1984; Zbl 0643.33001)] can compute the first zeros with an increasing accuracy. Finally, the author extends the available terms in the general asymptotic expansion [\textit{M. Abramowitz} and \textit{I. A. Stegun}, loc. cit., Section 9.10] which applies to the zeros of the Kelvin functions \(\text{ber}_n\), \(\text{bei}_n\), \(\text{ker}_n\), \(\text{kei}_n\), providing with a good numerical evidence. A recent paper by \textit{B. R. Fabijonas} and \textit{F. W. J. Olver} [SIAM Rev. 41, 762-773 (1999; Zbl 1053.33003)] does a similar task for the zeros of Airy functions.