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On the Riesz-Riemann-Liouville Integral

Published online by Cambridge University Press:  20 January 2009

E. T. Copson
E. T. Copson
Affiliation:
University College (Dundee), University of St Andrews.
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In a lecture at the Oslo Congress in 1936, Marcel Riesz introduced an important generalisation of the Riemann-Liouville integral of fractional order. Riesz's integral Iaf of order α is a multiple integral in m variables which converges uniformly when the real part of αexceeds m —2 and so represents an analytic function of the complex variable α. This integral is important in the theory of the generalised wave equation, for it provides a direct method of solving Cauchy's initial-value problem. The most recent developments show that it is likely to be also of great importance in quantum electrodynamics.

Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 8 , Issue 1 , October 1947 , pp. 25 - 36
Copyright
Copyright © Edinburgh Mathematical Society 1947

References

page 25 note 1 Comptes rendus du congrès international des mathématiciens (Oslo, 1936). Tome 2, pp. 44–45.Google Scholar

page 25 note 2 See a letter in Nature, 157, 734 (1946), by Gustafsou, T..CrossRefGoogle Scholar

page 26 note 1 Baker, and Copson, , The Mathematical Theory of Huygens' Principle (Oxford, 1939), pp. 60–61.Google ScholarCopson, , Proc. Boy. Soc. Edin. (A) 71, 260–272 (1943).Google Scholar

page 26 note 2 Kungl, . Fysiograjiska Scillskapets i Lund Forhandlingar, Bd. 15, Nr. 27 (1945).Google Scholar

page 28 note 1 Cf.Baker, and Copson, , loc. cit., p. 60, equation (7.41).Google Scholar

page 28 note 2 Cf.Fremberg, , loc. cit., p. 270.Google Scholar

page 30 note 1 If m = 2k + 3, there is a contribution from the lower limit θ=0, and this is important in § 6.

page 32 note 1 Fremberg, loc. cif., p. 274 I am grateful to a referee for pointing out that Fremberg's lemma, which omits reference to any“unspecified parameters”, is really insufficient. The unspecified parameters are to be the angle-variables φ1, φ2,… φn – 1.Google Scholar