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Application of the Mellin Inversion Theorem to Impulses

Published online by Cambridge University Press:  03 November 2016

N. W. McLachlan
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Extract

Introduction. Impulses occur not only in technical work but in everyday existence. There is hammering and door banging which may cause annoyance, handclapping which denotes enthusiasm and enjoyment, motoring on rough roads which causes discomfort (especially after a good meal!), speech, electrical communication in the Morse code, and so on ad infinitum. Impulses have been treated by applied mathematicians with reference to various technical problems, e.g. Heaviside in his work on electromagnetic theory. Moreover, the subject is one which merits attention, so in what follows a brief outline will be given of recent work based upon a particular case of the Mellin inversion theorem.

Type
Research Article
Information
The Mathematical Gazette , Volume 23 , Issue 255 , July 1939 , pp. 270 - 277
Copyright
Copyright © Mathematical Association 1939 

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References

page no 270 note * See McLachlan, , Complex Variable and Operational Calculus (Cambridge, 1939)Google Scholar, where a complete proof of the Mellin theorem and the conditions for validity are given in Appendix 4. In the Math. Gaz. 22, 264 (1938), Proffessor Carslaw has commented upon the presence of p outside an integral of type (1). The reason for the external p is twofold: (a) the operational forms (Laplace transforms) then obtained agree with those used by Heaviside, which are of long standing, (b) if t and p are considered to have dimensions d and d −1 respectively, so that the indent pt is dimensionless, f(t) and φ(p) in (1) have the same dimensions, which is useful for checking purposes. See Phil. Mag. 26, 394 (1938).

page no 271 note * Loc. cit.

page no 272 note * See Carslaw, Math. Gaz. loc. cit. for references.

page no 272 note † By virtue of this condition the integration is in effect from t = 0 to ∞.

page no 272 note ‡ The sign ⊃ means that φ(p)0/h is the operational form of ξ(t), in this case for the range t = 0 to h. It was introduced in Phil. Mag. 26, 394 (1938).

page no 273 note * From a list of O.F.S. if possible.

page no 273 note † It is not implied that p ≡ d/dt. This substitution gives φ 2(p) immediately and saves labour.

page no 275 note * This means that there is a time derivative on the l.h.s. of the differential equation, the driving force or disturbance being represented on the r.h.s. by (32).