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Some new asymptotic expansions for Bessel functions of large orders

Published online by Cambridge University Press:  24 October 2008

F. W. J. Olver
Affiliation:
National Physical LaboratoryTeddington, Middlesex

Extract

During the course of recent work (6) on the zeros of the Bessel functions Jn(x) and Yn(x), it became evident that the theory of the asymptotic expansion of Bessel functions whose arguments and orders are of comparable magnitudes was incomplete. The existing expansions for large orders are those of Debye and Meissel, detailed derivations of both of which are given by Watson ((8), pp. 237–48).

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1952

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References

(1)British Association Mathematical Tables, part-vol. B, The Airy integral (Cambridge, 1946).Google Scholar
(2)Copson, E. T.Theory of functions (Oxford, 1935), pp. 323–4.Google Scholar
(3)Imai, I.Asymptotic solutions of ordinary differential equations of the second order. Phys. Rev. (2), 80 (1950), 1112.CrossRefGoogle Scholar
(4)Langer, R. E.On the asymptotic solutions of ordinary differential equations with application to Bessel functions of large order. Trans. Amer. math. Soc. 33 (1931), 2364.CrossRefGoogle Scholar
(5)Nicholson, J. W.The asymptotic expansion of Bessel functions. Phil. Mag. (6), 19 (1910), 228–49.CrossRefGoogle Scholar
(6)Olver, F. W. J.A further method for the evaluation of zeros of Bessel functions and some new asymptotic expansions for zeros of functions of large order. Proc. Camb. phil. Soc. 47 (1951), 699712.CrossRefGoogle Scholar
(7)Watson, G. N.Bessel functions of large order. Proc. Camb. phil. Soc. 19 (1918), 96110.Google Scholar
(8)Watson, G. N.Theory of Bessel functions (Cambridge, 1944).Google Scholar
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