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On the Singular Behavior of the Inverse Laplace Transforms of the Functions $\frac{{{I}_{n}}\left( s \right)}{sI_{n}^{\prime }\left( s \right)}$

Published online by Cambridge University Press:  20 November 2018

Serguei Iakovlev*
Affiliation:
Department of Engineering Mathematics Dalhousie University Halifax, Nova Scotia B3J 2X4, e-mail: Serguei.Iakovlev@Dal.Ca
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Abstract

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Exact analytical expressions for the inverse Laplace transforms of the functions $\frac{{{I}_{n}}\left( s \right)}{sI_{n}^{\prime }\left( s \right)}$ are obtained in the form of trigonometric series. The convergence of the series is analyzed theoretically, and it is proven that those diverge on an infinite denumerable set of points. Therefore it is shown that the inverse transforms have an infinite number of singular points. This result, to the best of the author’s knowledge, is new, as the inverse transforms of $\frac{{{I}_{n}}\left( s \right)}{sI_{n}^{\prime }\left( s \right)}$ have previously been considered to be piecewise smooth and continuous. It is also found that the inverse transforms have an infinite number of points of finite discontinuity with different left- and right-side limits. The points of singularity and points of finite discontinuity alternate, and the sign of the infinity at the singular points also alternates depending on the order $n$. The behavior of the inverse transforms in the proximity of the singular points and the points of finite discontinuity is addressed as well.

Type
Research Article
Copyright
Copyright © Canadian Mathematical Society 2002

References

[1] Abramowitz, M. and Stegun, I. A., Handbook of Mathematical Functions. Dover, 1965.Google Scholar
[2] Bateman, H. and Erdelyi, A., Higher transcendental functions, Vol. 1. McGraw-Hill Book Company, New York, 1953.Google Scholar
[3] Bateman, H. and Erdelyi, A., Higher transcendental functions, Vol. 2. McGraw-Hill Book Company, New York, 1953.Google Scholar
[4] Geers, T. L., Excitation of an elastic cylindrical shell by a transient acoustic wave. Trans. ASME J. Appl. Mech. 36 (1969), 361969.Google Scholar
[5] Guz, A. N. and Kubenko, V. D., Theory of non-stationary aero- and hydroelasticity of shells. Naukova Dumka, Minsk, 1982, in Russian.Google Scholar
[6] Pertsev, A. K. and Platonov, E. G., Dynamics of shell and plates. Sudostroenie, Leningrad, 1987, in Russian.Google Scholar