The distribution composition \((x_+^r)^{-s}\) (Q5939789)

The authors study the distribution composition \((x^r_+)^{-s}\) and obtain as main theorem the interesting formula NEWLINE\[NEWLINE(x^r_+)^{-s}= x^{-rs}_+- (-1)^{rs} {(-1)^s[2s! C(\rho)+ \phi(s- 1)]+ rs\phi(rs- 1)\over (rs)!} \delta^{(rs- 1)}(x)NEWLINE\]NEWLINE for \(r,s= 1,2,3,\dots\)\ .NEWLINENEWLINENEWLINEThey prove this formula by applying the concepts of the neutrix limit and some ideas from \textit{I. M. Gel'fand} and \textit{G. E. Shilov} [``Generalized Functions, Vol. 1'', New York (1964; Zbl 0115.33101)].NEWLINENEWLINENEWLINEThey also extend the main theorem to the distribution \((x^r_+\pm i0)^{-s}\) and obtain the formula NEWLINE\[NEWLINE\begin{multlined} (x^r_+\pm i0)^{-s}= x^{-rs}_+- (-1)^{rs} {(-1)^s s![2C(\rho)+ \phi(s- 1)]+ rs\phi(rs- 1)\over (rs)!}\times\\ \delta^{(rs- 1)}(x)\pm{1\over 2}(-1)^{rs} {is\pi\over (rs)!} \delta^{(rs- 1)}(x)\end{multlined}NEWLINE\]NEWLINE for \(rs= 1,2,3,\dots\)\ .