On the Composition and Neutrix Composition of the Delta Function with the Hyperbolic Tangent and Its Inverse 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 $n=\mathrm{1,2},\dots$ and $\left\{{\delta }_n\left(x\right)\right\}$ is a certain regular sequence converging to the Dirac delta function. It is proved that the neutrix composition ${\delta }^{(rs-1)}\left(\right(\tanh x_{+}{)}^{1/r})$ exists and ${\delta }^{(rs-1)}\left({\left(\tanh x_{+}\right)}^{1/r}\right)={{\displaystyle \sum }}_{k=0}^{s-1}{{\displaystyle \sum }}_{i=0}^{K_k}\left({(-1)}^k{c}_{s-2i-1,k}\right(rs)!/2sk!){\delta }^{\left(k\right)}\left(x\right)$ for $r,s=\mathrm{1,2},\dots$, where $K_k$ is the integer part of $(s-k-1)/2$ and the constants $c_{j,k}$ are defined by the expansion ${\left({\tanh }^{-1}x\right)}^k={\left\{{\displaystyle \sum _{i=0}^{\infty }}(x^{2i+1}/(2i+1\left)\right)\right\}}^k={\displaystyle \sum _{j=k}^{\infty }}{c}_{j,k}{x}^j$, for $k=0,1,2,\dots$. Further results are also proved.