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On Certain Dual Integral Equations

Published online by Cambridge University Press:  18 May 2009

E. T. Copson
E. T. Copson
Affiliation:
St. Salvator's College, University of St. Andrews
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In his book on Fourier Integrals, Titchmarsh [l] gave the solution of the dual integral equations

for the case α > 0, by some difficult analysis involving the theory of Mellin transforms. Sneddon [2] has recently shown that, in the cases v = 0, α = ±½, the problem can be reduced to an Abel integral equation by making the substitution

or

It is the purpose of this note to show that the general case can be dealt with just as simply by putting

The analysis is formal: no attempt is made to supply details of rigour.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 5 , Issue 1 , January 1961 , pp. 21 - 24
Copyright
Copyright © Glasgow Mathematical Journal Trust 1961

References

1.Titchmarsh, E. C., Introduction to the theory of Fourier integrals (Clarendon Press, Oxford, 1937), pp. 334–339.Google Scholar
2.Sneddon, I. N., The elementary solution of dual integral equations, Proc. Glasgow Math. Assoc., 4 (1960), 108–110.CrossRefGoogle Scholar
3.Watson, G. N., A treatise on the theory of Bessel functions (University Press, Cambridge, 1944), p. 401, equations (1) and (3).Google Scholar
4.Whittaker, E. T. and Watson, G. N., A course of modern analysis (University Press, Cambridge, 1920), p. 229.Google Scholar