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Historical Note on Heaviside’s Operational Method

Published online by Cambridge University Press:  03 November 2016

N. W. McLachlan
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Extract

Operational methods in pure mathematics, e.g. the work of Boole, Graves and Murphy, existed many years before Oliver Heaviside introduced new ideas into mathematical physics. His first paper using such ideas was published in the Proceedings of the Royal Society in 1893. Remarks therein indicate his unfamiliarity with some of the literature of the subject, owing to lack of reference facilities, for example, Liouville’s work on fractional differentiation.

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Type
Research Article
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The Mathematical Gazette , Volume 22 , Issue 250 , July 1938 , pp. 255 - 260
Copyright
Copyright © Mathematical Association 1938

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References

page no 255 note * Proc. Roy. Soc. A. 52, 504 (1893).

page no 255 note † E. M. Theory, vol. 2, 34 (1922).

page no 255 note ‡ Proc. L.M.S. (2), 15, 412 (1916).

page no 255 note § Archiv für Elektrotechnik 4, Band 5, 159 (1916).

page no 255 note ‖ E.M.T. vol. 2, pp. 10, 11 (1922 edition). We disagree with Heaviside: see for example Math. Gaz. 22, 37, 1938, where a compact solution of a cable problem includes a definite integral which can be interpreted physically.

page no 255 note ** loc. cit.

page no 256 note * Electrical Papers, vol. 2, p. 259.

page no 256 note † Whittaker, E. T., “Oliver Heaviside”, Bull. Math. Soc. Calcutta, 20, 199 (1928-29)Google Scholar.

page no 256 note ‡ loc. cit.

page no 256 note § All the singularities of the integrand lie to the left of the contour. If they are all poles, c ± i ∞ may be replaced by a circle enclosing them.

page no 257 note * loc. cit. p. 412.

page no 257 note † The original was given to A. T. Starr when at Cambridge under Bromwich. The author is indebted to the former for a lantern slide of the original script.

page no 257 note ‡ Presumably the late Lord Rayleigh.

page no 257 note § loc. cit.

page no 258 note * See McLachlan, , Phil. Mag. 25, 261, 1938 CrossRefGoogle Scholar.

page no 258 note † Titchmarsh, Introduction to Theory of Fourier Integrals (1937).

page no 259 note * See McLachlan, Math. Gaz. loc. cit.

page no 259 note † As an example of this we can take the case of . The contour integral method yields , but the series for this would not be easily recognised. This occurs in a problem on elasto-viscosity of materials like flour-dough or gelatine solution.

page no 260 note * The most powerful method for solving technical problems of various kinds is that of integral equations. The Mellin theorem provides a ready method of solution for a certain class of problem.