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The Remainders in the Asymptotic Expansions of Certain Bessel Functions

Published online by Cambridge University Press:  24 October 2008

D. Burnett
D. Burnett
Affiliation:
Clare College
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Extract

The usual methods of investigation of asymptotic expansions of the various types of Bessel Functions show that the remainder is less in absolute value than the first term neglected. A more refined result was obtained by Stieltjes for K0(x) and certain other functions of order zero; he found an asymptotic series for the remainder and showed that the error due to stopping at one of the smallest terms is of the order of half the first term neglected. Watson notes that it would be of some interest to obtain corresponding results for functions of any order, and this is the object of this note. The method is quite different from that of Stieltjes.

Type
Articles
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 26 , Issue 2 , April 1930 , pp. 145 - 151
Copyright
Copyright © Cambridge Philosophical Society 1930

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References

* Cf. G. N. Watson, Theory of Bessel Functions, Chapter VII.

† Watson, pp. 213, 214.

‡ Results for functions whose order is between −½ and +½ have recently been given by Koshliakov, , Journal London Math. Soc. 4 (1929), p. 297; his method is an extension of Stieltjes'. I am indebted to a referee for drawing my attention to this paper.CrossRefGoogle Scholar

§ Watson, p. 207.

* Watson, p. 196.