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XXII.—Some Applications of Marcel Riesz's Integrals of Fractional Order

Published online by Cambridge University Press:  14 February 2012

E. T. Copson
E. T. Copson
Affiliation:
University College, Dundee, in the University of St Andrews
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Summary

An account is given of Professor Marcel Riesz's generalisation of the Riemann-Liouville integral of fractional order. It is shown that the new ideas introduced by Riesz may prove valuable in the theory of partial differential equations and in the theory of the wave-equation in momentum space.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 61 , Issue 3 , 1943 , pp. 260 - 272
Copyright
Copyright © Royal Society of Edinburgh 1943

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References

page 260 note * Comptes rendus du congrès international des mathématiciens (Oslo, 1936), vol. ii, pp. 44-45. 62–63.Google Scholar

page 260 note † See, for example, , Hadamard, Lectures on Cauchy's Problem (Yale, 1923)Google Scholar; , Courant and , Hilbert, Methoden des mathematischen Physik (Berlin, 1938), vol. ii, p. 438Google Scholar.

page 260 note ‡ “Potentiel d'Equilibre et Capacité des Ensembles,” Medd. Lunds Univ. mat. Sent. (1935), vol. iii.Google Scholar

page 260 note § , Baker and , Copson, The Mathematical Theory of Huygens' Principle (Oxford, 1939), pp. 54–67.Google Scholar

page 264 note * Lemma 3 is rather difficult to prove. The simplest method seems to be to apply a Lorentz transformation which turns the integral into the same expression with X, Y, Z replaced by zero and T2 by T2-X2- Y2-Z2.

page 265 note * Proc. Hoy. Soc, A, vol. clxxii, 1939, pp. 384-409.

page 265 note † See, for example, , Courant-Hilbert, loc. cit., vol. ii, p. 443Google Scholar.

page 268 note * Physical units have been chosen so that h/2πin is unity.

page 268 note † My thanks are due to Dr C. A. Coulson for suggesting this problem to me, and for his kind and valuable criticism of this work.

page 269 note * Zeits. Phys., vol. xcviii, 1935, pp. 145–154.Google Scholar

page 269 note † This proof of the formula for the operator 1/r was suggested to me by Professor E. T. Whittaker. It is added here (January 1943) at the request of a referee.

page 271 note * It has been pointed out by a referee that the operational interpretation of i/r12 can also be obtained by the use of Fourier's integral by an argument similar to that given at the end of § 6. The relation where integration is through the whole of p-space, is needed in the course of the proof.