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An integral equation

Published online by Cambridge University Press:  24 October 2008

G. H. Hardy  and
E. C. Titchmarsh
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Extract

1. The integral equation

where x, f (x), and λ are real and α positive, may be regarded as a differential equation of order α. Suppose for example that α is a positive integer p, that f (x) tends to 0, when x → ∞, with sufficient rapidity, and that

Then, if we integrate repeatedly by parts, and write z for fp (x), (1·1) becomes

The only solutions are finite combinations of exponentials.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 28 , Issue 2 , April 1932 , pp. 165 - 173
Copyright
Copyright © Cambridge Philosophical Society 1932

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References

REFERENCES

1.Bochner, S., ‘Über eine Klasse singulärer Integralgleichungen’, Berliner Sitzungsberichte, 22 (1930), 403–411.Google Scholar
2.Bosanquet, L. S., ‘On Abel's integral equation and fractional integrals’, Proc. London Math. Soc. (2), 31 (1930), 134–143.CrossRefGoogle Scholar
3.Hardy, G. H., ‘On Mellin's inversion formula’, Messenger of Math., 47 (1918), 178–184.Google Scholar
4.Hardy, G. H. and Titchmarsh, E. C., ‘Solution of certain integral equations considered by Bateman, Kapteyn, Littlewood and Milne’, Proc. London Math. Soc. (2), 23 (1924), 1–26. See also a supplementary paper, ibid., 30 (1930), 95–106.Google Scholar