Hostname: page-component-586b7cd67f-rdxmf Total loading time: 0 Render date: 2024-11-22T08:37:24.895Z Has data issue: false hasContentIssue false
  • English
  • Français

Laplace transforms for superexponential functions

Published online by Cambridge University Press:  17 February 2009

Michael A. B. Deakin
Michael A. B. Deakin
Affiliation:
Department of Mathematics, Monash University, Clayton, Victoria 3168, Australia
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

It is usually stated that the Laplace transform cannot be applied to most superexponential functions; indeed this limitation is sometimes represented as a deficiency of the technique. It is shown here that a generalisation serves to overcome much of the force of this objection. The generalisation is based on one first proposed over fifty years ago, but which is not widely known, nor was it ever worked out in detail.

Type
Research Article
Information
The ANZIAM Journal , Volume 36 , Issue 1 , July 1994 , pp. 17 - 25
Copyright
Copyright © Australian Mathematical Society 1994

References

[1]Abramowitz, M. and Stegun, I. A. (eds.), Handbook of mathematical functions (Dover Publications, New York, 1968).Google Scholar
[2]Berg, L., Introduction to the operational calculus (North Holland Publishing Co., Amsterdam, 1967).Google Scholar
[3]Erdélyi, A. (ed.), Higher trancendental functions. Volume I (McGraw Hill Book Co., New York, 1953).Google Scholar
[4]Erdélyi, A. (ed.), Tables of integral tranforms, Volume I (McGraw Hill Book Co., New York, 1954).Google Scholar
[5]Flajolet, P. and Schott, R., “Non-overlapping partitions, continued fractions, Bessel functions and a divergent series”, Europ. J. Combinatorics 11 (1990) 421–432.CrossRefGoogle Scholar
[6]Inayat-Hussein, A. A. and Buckingham, M. J., “On the Laplace transform of the G function”, J. Phys. A: Math. Gen. 19 (1986) L875–L876.CrossRefGoogle Scholar
[7]Krabbe, G., Operational calculus (Springer-Verlag, Berlin, 1970).CrossRefGoogle Scholar
[8]Liverman, T. P. G., Generalized functions and direct operational methods (Prentice-Hall, Englewood Cliffs, N. J., 1964).Google Scholar
[9]Mikusiński, J., “Le calcul opérationnel d'intervalle fini”, Studia Mathematica 15 (1956) 225–251.CrossRefGoogle Scholar
[10]Oliver, F. W. J., Asymptotics and special functions (Academioc Press New York, 1974).Google Scholar
[11]Rothman, M., “Tables of the integrals and differential coefficients of Gi (+x) and Hi (−x)”, Quart.J. Mech. Appl. Math. (1954) 379–384.CrossRefGoogle Scholar
[12]Spanier, J. and Oldham, K. B., An atlas of functions (Hemisphere Publishing Corp., Washington, 1987).Google Scholar
[13]Vignaux, J. C., “Sugli integrali di Laplace asintotici”, Atti Accad. naz. Lincei, Rend. CI. Sci.fis.mat. 29 (6 ser.) (1939) 396–402.Google Scholar
[14]Výborný, R., “Operational calculus, Laplace transforms or an algebraic approach”, Aust. Math. Soc.Gaz. 18 (1991) 71–78.Google Scholar
[15]Watson, G. N., “A theory of asymptotic series”, Proc. Roy. Soc. All (1911) 279–313.Google Scholar