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Zbl 0998.33005
MacLeod, Allan J.
Asymptotic expansions for the zeros of certain special functions. (English)
[J] J. Comput. Appl. Math. 145, No.2, 261-267 (2002). ISSN 0377-0427

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.

MSC 2000:: *33C10 Cylinder functions, etc.

Citations: Zbl 0303.41035; Zbl 0643.33001; Zbl 1053.33003

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