Generalization of distributional product of Dirac's delta in hypercone

Abstract

Let G=G(m, x) be defined by [image omitted] The hypersurface G is due to Kanathai and Nonlaopon ([Kananthai, A. and Nonlaopon, K., 2003, On the residue of generalized function P. Thai Journal of Mathematics, 1, 49-57]). We observe that putting m=1 we obtain [image omitted] The quadratic form P is due to Gelfand and Shilov [Gelfand, I.M. and Shilov, G.E., 1964, Generalized Function, Vol. 1 (New York: Academic Press), p. 253]. The hypersurface P=0 is a hypercone with a singular point (the vertex) at the origin. We know that the kth derivative of Dirac's delta in G there exists under conditions depending on n and m, where n is the dimension of the space. In our study, the main purpose is to related distribution product of the Dirac delta with the coefficient corresponding to the double pole of the expansion in the Laurent series of G+, where G is defined by (3). From this we can arrive at a formula in terms of the operator Lm which is defined by (16). Our results are generalizations of formulae that appear in Aguirre [Aguirre, T.M.A., 2000, The distributional product of Dirac's delta in a hypercone. Journal of Computation and Applied Mathematics, 115, 13-21], pp. 20-21.