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On the Heaviside operational calculus*

Published online by Cambridge University Press:  24 October 2008

Harold Jeffreys  and
D. P. Dalzell
Harold Jeffreys
Affiliation:
St John's CollegeCambridge
D. P. Dalzell
Affiliation:
61 Westbury Court London, S.W.4
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Extract

Some theorems in the operational calculus, taking definite integration as the fundamental operator, are proved for discrete systems. It is suggested that it is physically more satisfactory to regard the solution for a continuous system as the limit of the solutions for a set of discrete systems, rather than as the solution of a partial differential equation. The necessary and sufficient condition for the validity in this sense of the usual operational method for continuous systems appears to be that the discrete systems considered shall not tend to infinite instability; that is, that all poles of the Bromwich integrand shall always have real parts less than some fixed positive quantity. The correction for finiteness of the number of degrees of freedom is examined for a case analogous to heat conduction and found to be unimportant.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 36 , Issue 3 , July 1940 , pp. 267 - 282
Copyright
Copyright © Cambridge Philosophical Society 1940

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References

† Proc. London Math. Soc. 15 (1914), 401–48.Google Scholar

‡ Dalzell, D. P., Proc. Phys. Soc. 42 (1929), 75–81CrossRefGoogle Scholar; van der Pol, B., Phil. Mag. 7 (1929), 1153–62CrossRefGoogle Scholar; Carslaw, H. S., Math. Gaz. 22 (1938), 264–80CrossRefGoogle Scholar; McLachlan, N. W., Math. Gaz. 23 (1939), 270.CrossRefGoogle Scholar

* Electric circuit theory and the operational calculus (1926).Google Scholar

† Operational methods in mathematical physics (Cambridge, 1927 and 1931).Google Scholar

* Jeffreys, , Proc. Cambridge Phil. Soc. 23 (1927), 768–78.CrossRefGoogle Scholar

* Conversely, in the operational derivation of interpolation formulae, e nD does mean application of Taylor's theorem, and commutes with D. Hence correct results are obtained for derivatives, but wrong ones are sometimes found for integrals if 1/D is identified with integration.

* Titchmarsh, , Theory of functions (Oxford, 1932), p. 168.Google Scholar

† Cf. Titchmarsh, loc. cit. p. 438. The form used here is an immediate corollary of that in the book.

* Titchmarsh, loc. cit. p. 387.

* Operational methods, p. 45, equation, (9).