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Asymptotic expansions for the zeros of certain special functions. (English) Zbl 0998.33005
In this paper asymptotic formulae for the zeros of the cosine-integral $Ci\left(x\right)$, the Struve function ${H}_{0}\left(x\right)$ as well as the Kelvin functions are derived showing an acceptable degree of accuracy. By using the standard technique by F. W. J. Olver [Asymptotics and special functions, Chapter 1, Academic Press, New York (1974; Zbl 0303.41035)] the obtained expression for $Ci\left(x\right)$ gives more than 10D accuracy for all roots beyond the ninth. The corresponding expansion to ${H}_{0}\left(x\right)$ [see M. Abramowitz and 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 [M. Abramowitz and 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 B. R. Fabijonas and F. W. J. Olver [SIAM Rev. 41, 762-773 (1999; Zbl 1053.33003)] does a similar task for the zeros of Airy functions.
##### MSC:
 33C10 Bessel and Airy functions, cylinder functions, ${}_{0}{F}_{1}$