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Published online by Cambridge University Press: 24 October 2008
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.
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† 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).
