Regularization and Distributional Derivatives of (x^2_1 + x^2_2 + ldots + x^2_p)^{1/2n} in R^p
Abstract
Our main aim is to present the value of the distributional derivative {partial}^N/partial x^{k_1_1 partial x^{k_2}_2 \cdots partial x^{k_p}_p}(1/r^n), where r = (x^2_1+x^2_2+ \cdots +x^2_p)^{1/2} in R^p, N = k_1 + k_2 + \cdots + k_p, and p, n, k_1, k_2, \cdots, k_p are positive integers. For this purpose, we first define a regularization of 1/x^n in R^1, which in turn helps us to define the regularization of 1/r^n in R^p. These regularizations are achieved as asymptotic limits of the truncated functions H(xɛ)/x^n and H(rɛ)/r^n as ɛ > 0, plus certain terms concentrated at the origin, where H is the Heaviside function. In the process of the derivation of the distributional derivative formula mentioned, we also derive many other interesting results and introduce some simplifying notation.
 Publication:

Proceedings of the Royal Society of London Series A
 Pub Date:
 October 1985
 DOI:
 10.1098/rspa.1985.0099
 Bibcode:
 1985RSPSA.401..281E