Hadamard’s finite part concept in dimension <i>n</i> ≥ 2, distributional definition, regularization forms and distributional derivatives

Abstract

Abstract The aim of this paper is to apply the Hadamard finite part concept to singular pseudo-functions such as ɑ(θ)rβlogj r, where r2 : = x12 +. . . + xn2 in Rn, j ϵ N, β ≤ -n, and a is a function depending on θ : = OM/r. For these specific pseudo­-functions both regularizations forms and distributional derivatives, ∂p∂x1p1...∂x1pn[α(θrβlogj⁡r)], where p, p1, . . . , pn are positive integers with p= p1+ . . . + pn are investigated. Whenever possible the results are compared with those obtained by Estrada &amp; Kanwal (1985, 1989).