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Regularization and distributional derivatives of \((x^ 2_ 1+x^ 2_ 2+\dots +X^ 2_ p)^{-n}\) in \({\mathbb{R}}^ p\). (English) Zbl 0578.46034

Summary: Our main aim is to present the value of the distributional derivative \[ \frac{{\bar \partial}^ N}{\partial x_ 1^{k_ 1}\partial x_ 2^{k_ 2}...\partial x_ p^{k_ p}}(\frac\quad {1}{r^ n}), \] where \(r=(x^ 2_ 1+x^ 2_ 2+...+x^ 2_ p)^{1/2}\) in \({\mathbb{R}}^ p\), \(N=k_ 1+k_ 2+...+k_ p\), and \(p,n,k_ 1,k_ 2,...,k_ p\) are positive integers. For this purpose, we first define a regularization of \(1/x^ n\) in \({\mathbb{R}}^ 1\), which in turn helps us to define the regularization of \(1/r^ n\) in \({\mathbb{R}}^ p\). These regularizations are achieved as asymptotic limits of the truncated functions \(H(x-\epsilon)/x^ n\) and \(H(r-\epsilon)/r^ n\) as \(\epsilon\) \(\to 0\), 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.

MSC:

46F10 Operations with distributions and generalized functions
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