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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 $Ci(x)$, 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 {\it 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 {\it M. Abramowitz} and {\it 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 [{\it M. Abramowitz} and {\it 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 {\it B. R. Fabijonas} and {\it F. W. J. Olver} [SIAM Rev. 41, 762-773 (1999; Zbl 1053.33003)] does a similar task for the zeros of Airy functions.
33C10Bessel and Airy functions, cylinder functions, ${}_0F_1$
Full Text: DOI
[1] Abramowitz, M.; Stegun, I. A.: Handbook of mathematical functions. (1965) · Zbl 0171.38503
[2] Fabijonas, B. R.; Olver, F. W. J.: On the reversion of an asymptotic expansion and the zeros of the Airy functions. SIAM rev. 41, 762-773 (1999) · Zbl 1053.33003
[3] Olver, F. W. J.: Asymptotics and special functions. (1974) · Zbl 0303.41035