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A Class of Integral Transforms

  • Jet Wimp (a1)
Abstract

In this paper we discuss a new class of integral transforms and their inversion formula. The kernel in the transform is a G-function (for a treatment of this function, see ((1), 5.3) and integration is performed with respect to the argument of that function. In the inversion formula, the kernel is likewise a G-function, but there integration is performed with respect to a parameter. Known special cases of our results are the Kontorovitch-Lebedev transform pair ((2), v. 2; (3))

and the generalised Mehler transform pair (7)

These transforms are used in solving certain boundary value problems of the wave or heat conduction equation involving wedge or conically-shaped boundaries, and are extensively tabulated in (6).

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Copyright
References
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(1) Erdélyi A., Magnus W., Oberhettinger F. and Tricomi F. G., Higher Transcendental Functions, 3 vols. (McGraw-Hill, 1953).
(2) Erdélyi A., Magnus W., Oberhettinger F. and Tricomi F. G., Tables of Integral Transforms, 2 vols. (McGraw-Hill, 1954).
(3) Kontorovich M. J. and Lebedev N. N., On a method of solution of some problems in diffraction theory, J. Phys. Moscow, 1 (1939), 229241.
(4) Magnus W. and Oberhettinger F., Formeln und Sätze für die Speziellen Funktionen der Mathematischen Physik, 2nd ed. (Springer-Verlag, Berlin, 1948), p. 189.
(5) Mercer A. McD., Integral transform pairs arising from second order differential equations, Proc. Edin. Math. Soc. (2), 13 (1962), 6368.
(6) Oberhettinger F. and Higgins T. P., Tables of Lebedev, Mehler and generalised Mehler transforms, Math. Note No. 246, Boeing Scientific Research Laboratories.
(7) Rosenthal P. L., On a Generalization of Mehler's Inversion Formula and Some of its Applications, dissertation (Oregon State University, 1961).
(8) Slater L. J., Confluent Hypergeometric Functions (Cambridge University Press, 1960).
(9) Titchmarsh E. C., Eigenfunction Expansions Associated with Second-Order Differential Equations, 2nd ed. (Oxford, 1962).
(10) Titchmarsh E. C., Introduction to the Theory of Fourier Integrals, 2nd ed. (Oxford, 1948).
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Proceedings of the Edinburgh Mathematical Society
  • ISSN: 0013-0915
  • EISSN: 1464-3839
  • URL: /core/journals/proceedings-of-the-edinburgh-mathematical-society
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