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A property of the zeros of a cross-product of Bessel functions

Published online by Cambridge University Press:  24 October 2008

D. M. Willis
D. M. Willis
Affiliation:
D.S.I.R. Radio Research Station, Ditton Park, Slough
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Extract

In this note it is shown that any positive root of the transcendental equation

is definable as a continuous increasing function of the real variable ν, provided ν is positive. Here Jν and Yν denote respectively the Bessel functions of the first and second kind of order ν, and k is a positive constant. Watson ((4)) has established the corresponding result for the simpler equations Jν(z) = 0 and Jν(z) cosα − Yν(z) sinα = 0, where α is a constant. The extension of the result to the positive roots of equation (1) is important because this equation occurs quite frequently in physical problems.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 61 , Issue 2 , April 1965 , pp. 425 - 428
Copyright
Copyright © Cambridge Philosophical Society 1965

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References

REFERENCES

(1)Fletcher, A., Miller, J. C. P., Rosenhead, L. and Comrie, L. J.An index of mathematical tables, vol. 1 (Scientific Computing Service; London, 1962).Google Scholar
(2)Gray, A., Mathews, G. B. and Macrobert, T. M.A treatise on Bessel functions and their applications to physics (Macmillan; London, 1922).Google Scholar
(3)McMahon, J.Ann. of Math. 9 (1895), 23–30.CrossRefGoogle Scholar
(4)Watson, G. N.A treatise on the theory of Bessel functions (2nd ed.; Cambridge, 1944).Google Scholar