On the Composition and Neutrix Composition of the Delta Function and the Function cosh^{-1}(|x|^{1/r}+1)

Abstract

Let $F$ be a distribution in $\mathcal{D'}$ and let $f$ be a locally summable function. The composition $F(f(x))$ of $F$ and $f$ is said to exist and be equal to the distribution $h(x)$ if the limit of the sequence $\{ F_{n}(f(x))\}$ is equal to $h(x)$, where $F_n(x) =F(x)*\delta _n(x)$ for $n=1,2, \ldots$ and $\{\delta_n(x)\}$ is a certain regular sequence converging to the Dirac delta function. It is proved that the neutrix composition $ \delta^{(s)}[\cosh^{-1} (x_+^{1/r}+1)] $ exists and \beqa \delta^{(s)}[\cosh^{-1} (x_+^{1/r}+1)] = - \sum _{k=0} ^{M-1} \sum_{i=0}^{kr+r} {k \choose i}{(-1)^{i+k}rc_{r,s,k} \over (kr+r)k!}\delta ^{(k)}(x), for $s =M-1,M, M+1,\ldots$ and $r=1,2,\ldots,$ where $$c_{r,s,k}=\sum _{j=0}^{i} {i \choose j}{ (-1)^{kr+r-i}(2j-i)^{s+1}\over 2^{s+i+1} },$$ $M$ is the smallest integer for which $s-2r+1 < 2Mr$ and $r\le s/(2M+2).$ Further results are also proved.