## Abstract

Let $F$ be a distribution in $\mathcal{D}\text{'}$ 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 ${\delta}^{(rs-1)}\left(\right(\mathrm{tanh}{x}_{+}{)}^{1/r})$ exists and ${\delta}^{(rs-1)}\left({\left(\mathrm{tanh}{x}_{+}\right)}^{1/r}\right)={{\displaystyle \sum}}_{k=0}^{s-1}{{\displaystyle \sum}}_{i=0}^{{K}_{k}}\left({(-1)}^{k}{c}_{s-2i-1,k}\right(rs)!/2sk!){\delta}^{\left(k\right)}\left(x\right)$ 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 ${\left({\mathrm{tanh}}^{-1}x\right)}^{k}={\left\{{\displaystyle \sum _{i=0}^{\infty}}({x}^{2i+1}/(2i+1\left)\right)\right\}}^{k}={\displaystyle \sum _{j=k}^{\infty}}{c}_{j,k}{x}^{j}$, for $k=0,1,2,\dots $. Further results are also proved.

## Citation

Brian Fisher. Adem Kılıçman. "On the Composition and Neutrix Composition of the Delta Function with the Hyperbolic Tangent and Its Inverse Functions." J. Appl. Math. 2011 1 - 13, 2011. https://doi.org/10.1155/2011/846736

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