Regularization and Distributional Derivatives of $(x^2_1 + x^2_2 + \ldots + x^2_p)^{-\frac{1}{2}n}$ in $\mathbb{R}^p$

R. Estrada, R. P. Kanwal

Abstract

Our main aim is to present the value of the distributional derivative $\frac{\overline{\partial}^N}{\partial x^{k_1}_1 \partial x^{k_2}_2 \cdots \partial x^{k_p}_p}(\frac{1}{r^n})$, where $r = (x^2_1+x^2_2+ \cdots +x^2_p)^{\frac{1}{2}}$ in $\mathbb{R}^p, N = k_1 + k_2 + \cdots + k_p$, 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(x-$\epsilon$)/x$^n$ and H(r-$\epsilon$)/r$^n$ as $\epsilon \rightarrow 0,$ 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.

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