On the Neutrix Composition of the Delta and Inverse Hyperbolic Sine Functions

Abstract

Let $F$ be a distribution in $𝒟\text{'}$ and let $f$ be a locally summable function. The composition $F\left(f\right(x\left)\right)$ of $F$ and $f$ is said to exist and be equal to the distribution $h\left(x\right)$ if the limit of the sequence $\left\{F_n\left(f\right(x\left)\right)\right\}$ is equal to $h\left(x\right)$, where $F_n\left(x\right)=F\left(x\right)*{\delta }_n\left(x\right)$ for $n=1,2,\dots$ and $\left\{{\delta }_n\left(x\right)\right\}$ is a certain regular sequence converging to the Dirac delta function. In the ordinary sense, the composition ${\delta }^{\left(s\right)}\left[\right({\sinh }^{-1}{x}_{+}{)}^r]$ does not exists. In this study, it is proved that the neutrix composition ${\delta }^{\left(s\right)}\left[\right({\sinh }^{-1}{x}_{+}{)}^r]$ exists and is given by ${\delta }^{\left(s\right)}\left[{\left({\sinh }^{-1}{x}_{+}\right)}^r\right]={\displaystyle \sum _{k=0}^{sr+r-1}}{\displaystyle \sum _{i=0}^k}\left(\genfrac{}{}{0ex}{}{\text{k}}{\text{i}}\right)({(-1)}^{k}r{c}_{s,k,i}/2^{k+1}k!){\delta }^{\left(k\right)}\left(x\right)$, for $s=0,1,2,\dots$ and $s=1,2,\dots$, where $c_{s,k,i}=(-1{)}^{s}s!\left[\right(k-2i+1{)}^{rs-1}+(k-2i-1{)}^{rs+r-1}]/\left(2\right(rs+r-1)!)$. Further results are also proved.