On Powers Of The Heaviside Function For Negative Integers

Abstract

The distributions (x(+)(r))(-s) and (x(+)(r))_(-s) were defined as the neutrix limit of the sequences ((x(+)(r))(-s))n and ((x(+)(r))_(-s))n respectively for r, s = 1, 2,..., see [J.D. Nicholos, B. Fisher, The distribution composition (x(+)(r))(-s), J. Math. Anal. Appl. 258 (2001) 131-145; B. Fisher, On defining the distribution (x(+)(r))_(-s), Univ. u Novom Sadu Zb. Rad. Prirod. Mat. Fak. Ser. Mat. 15 (1985) 119-129]. We here consider these distributions when r = 0. In other words, we define the sth powers of the Heaviside function H(x) in the distributional sense for negative integers. Further compositions are also considered. (c) 2006 Elsevier Inc. All rights reserved.