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On Liouville's extension of Abel's integral equation

Published online by Cambridge University Press:  26 February 2010

L. S. Bosanquet
L. S. Bosanquet
Affiliation:
University College, London
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Extract

Necessary and sufficient conditions for Abel's integral equation to have a solution have been given by Tamarkin [18]. The form of the solution was obtained by Abel [1]. The corresponding integral equation for an infinite range of integration was introduced by Liouville [12], who found a solution in a restricted class of cases. In the present paper, we find necessary and sufficient conditions for Liouville's equation to have a solution, and also give the form of the solution.

Type
Research Article
Information
Mathematika , Volume 16 , Issue 1 , June 1969 , pp. 59 - 85
Copyright
Copyright © University College London 1969

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References

1.Abel, N. H., “Auflösung einer mechanischen Aufgabe”, J. für Math., 1 (1826), 153–157.Google Scholar
2.Andersen, A. F., “Summation af ikke hel Orden”, Mat. Tidsskrift, B (1946), 33–52.Google Scholar
3.Bosanquet, L. S., “On Abel's integral equation and fractional integrals”, Proc. London Math. Soc. (2), 31 (1931), 134–143.Google Scholar
4.Bosanquet, L. S., “On the order of magnitude of fractional differences”, Calcutta Math. Soc. Golden Jubilee Memorial Volume (1958–1959), 161–172.Google Scholar
5.Bosanquet, L. S., “Some extensions of M. Riesz's mean value theorem”, Indian J. of Math., 9 (1967), 65–90.Google Scholar
6.Cossar, H., “A theorem on Cesaro summability”, J. London Math. Soc., 16 (1941), 56–68.CrossRefGoogle Scholar
7.Hardy, G. H. and Littlewood, J. E., “Some properties of fractional integrals (I)”, Math. Zeit, 27 (1928), 565–606.CrossRefGoogle Scholar
8.Hardy, G. H. and Riesz, M., The general theory of Dirichkf s series (Cambridge Tract No. 18, 1915; reprinted 1952).Google Scholar
9.Isaacs, G. L., “M. Riesz's mean value theorem for infinite integrals”, J. London Math. Soc., 28 (1953), 171–176.CrossRefGoogle Scholar
10.Liouville, J., “Sur quelques questions de géométrie et de mechanique, et sur un nouveau genre de calcul pour resoudre ces questions”, J. de l'Ecole polytechnique, 13 (1832), 1–69.Google Scholar
11.Liouville, J., “Sur le calcul des differentielles á indices quelconques”, J. de l'Ecole polytechnique, 13 (1832), 71–162.Google Scholar
12.Liouville, J., “Memoire sur une formule d'analyse”, J. für Math., 12 (1834), 273–287.Google Scholar
13.Maddox, I. J., “A note on summability factor theorems”, Quart. J. of Math. (Oxford 2nd series), 15 (1964), 208–216.CrossRefGoogle Scholar
14.Riemann, B., “Versuch einer allgemeinen Auffassung der Integration und Differentiation”, Gesammelte Math. Werke (XIX) (2nd ed., 1892; Dover Publications 1953), 353–366.Google Scholar
15.Riesz, M., “Une méthode de sommation equivalente à la méthode des moyennes arithmétiques”, Comptes rendus, 152 (1911), 1651–54.Google Scholar
16.Riesz, M., “Sur un theoreme de la moyenne et ses applications”, Acta Liu. ac. Sci. Univ. Hungaricae {Szeged), 1 (1923), 114–126.Google Scholar
17.Saks, S., Theory of the integral (Warsaw-Lwow, French ed. 1933; English ed. 1937).Google Scholar
18.Tamarkin, J. D., “On integrable solutions of Abel's integral equation”, Annals of Math. (2), 31 (1930), 219–229.CrossRefGoogle Scholar
19.Weyl, H., “Bemerkungen zum Begriff des Differentialquotienten gebrochener Ordnung”, Vierteljahrsschrift d. naturf. Ges. in Zurich, 62 (1917), 296–302.Google Scholar