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On a singular eigenvalue problem

Published online by Cambridge University Press:  24 October 2008

D. Naylor  and
S. C. R. Dennis
D. Naylor
Affiliation:
University of Western Ontario
S. C. R. Dennis
Affiliation:
University of Western Ontario
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Extract

Sears and Titchmarsh (1) have formulated an expansion in eigenfunctions which requires a knowledge of the s-zeros of the equation

Here ka > 0 is supposed given and β is a real constant such that 0 ≤ β < π. The above equation is encountered when one seeks the eigenfunctions of the differential equation

on the interval 0 < α ≤ r < ∞ subject to the condition of vanishing at r = α. Solutions of (2) are the Bessel functions J±is(kr) and every solution w of (2) is such that r−½w(r) belongs to L2 (α, ∞). Since the problem is of the limit circle type at infinity it is necessary to prescribe a suitable asymptotic condition there to make the eigenfunctions determinate. In the present instance this condition is

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 64 , Issue 2 , April 1968 , pp. 439 - 446
Copyright
Copyright © Cambridge Philosophical Society 1968

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References

REFERENCES

(1)Sears, D. B. and Titchmarsh, E. CSome eigenfunction formulae. Quart. J. Math. Oxford (Series 2), 1 (1950), 165–175.CrossRefGoogle Scholar
(2)Naylor, D.On a Sturm-Liouville expansion in series of Bessel functions. Proc. Cambridge Philos. Soc. 62 (1966), 61–72.CrossRefGoogle Scholar
(3)Watson, G. N.A treatise on the theory of Bessel functions, 2nd edition (Cambridge, 1944).Google Scholar
(4)Cochran, J. A.The analyticity of cross product Bessel function zeros. Proc. Cambridge Philos. Soc. 62 (1966), 215–226.CrossRefGoogle Scholar
(5)Willis, D. M.A property of the zeros of a cross product of Bessel functions. Proc. Cambridge Philos. Soc. 61 (1965), 425–428.CrossRefGoogle Scholar
(6)Olver, F. W. J.Royal Society Mathematical Tables, vol. 7, Bessel functions, Part 3, Zeros and associated values (Cambridge University Press, 1960).Google Scholar
(7)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. Cambridge Philos. Soc. 47 (1951), 699–712.CrossRefGoogle Scholar
(8)Hildebrand, F. B.Introduction to numerical analysis, pp. 233–238. (McGraw-Hill, 1956).Google Scholar
(9)Erdélyi, A., Magnus, W., Oberhettinger, F. & Tricomi, F. G.Higher transcendental functions, vol. 2 (McGraw-Hill, 1953).Google Scholar