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Two index laws for fractional integrals and derivatives

Published online by Cambridge University Press:  09 April 2009

E. R. Love
E. R. Love
Affiliation:
Department of MathematicsThe University of MelbourneVictoria, 3052 Australia
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Summary

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The first index law, or addition theorem, is well known. The second is much less well known; but both have been found to be of importance in recent studies of hypergeometric integral equations. The first law has usually been considered only in the simple case of orders of integration which have positive real part, or in the context of generalized functions. Arising out of the need to manipulate expressions involving several fractional integrals and derivatives, our aim here is to establish both laws for all combinations of complex orders of integration and differentiation, and for nearly all functions for which the fractional derivatives involved exist as locally integrable functions.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 14 , Issue 4 , December 1972 , pp. 385 - 410
Copyright
Copyright © Australian Mathematical Society 1972

References

[1]Widder, D. V., ‘The Stieltjes Transform’, Trans. American Math. Soc. 43 (1938), 7–60.CrossRefGoogle Scholar
[2]Love, E. R. and Young, L. C., ‘On fractional integration by parts’, Proc. London Math. Soc. (2) 44 (1938), 1–35.CrossRefGoogle Scholar
[3]Hille, E., ‘Notes on linear transformations: II Analyticity of semigroups’, Annals of Math. (2) 40 (1939), 1–47.CrossRefGoogle Scholar
[4]Kober, H., ‘On fractional integrals and derivative’, Quart. J. Math. (Oxford) XI (1940) 193–211.Google Scholar
[5]Kober, H., ‘On a theorem of Schur and on fractional integrals of purely imaginary order’, Trans. American Math. Soc. 50 (1941), 160–174.Google Scholar
[6]Riesz, M., ‘L'intégrale de Riemann-Liouville et le problème de Cauchy’, Acta Math. 81 (1949), 1–223.CrossRefGoogle Scholar
[7]Erdélyi, A. et al. , Higher Transcendental Functions, vol. 1 (McGraw Hill, 1953).Google Scholar
[8]Gelfand, I. M. and Shilov, G. F., Generalized Functions, vol. 1 (Academic Press, 1963).Google Scholar
[9]Higgins, T. P., The Rodrigues Operator Transform, (Boeing Scientific Research Laboratories, Math. Note No. 437 (1965) 1–125.)CrossRefGoogle Scholar
[10]Erdélyi, A., ‘Some integral equations involving finite parts of divergent integrals’, Glasgow Math. J. 8 (1967), 50–54.CrossRefGoogle Scholar
[11]Love, E. R., ‘Some integral equations involving hypergeometric functions’, Proc. Edinburgh Math. Soc. (11) 15 (1967), 169–198.CrossRefGoogle Scholar
[12]Love, E. R., ‘Fractional derivatives of imaginary order’, J. London Math. Soc. (2) 3 (1971) 241–259.CrossRefGoogle Scholar