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2011 On the Neutrix Composition of the Delta and Inverse Hyperbolic Sine Functions
J. Appl. Math. 2011: 1-12 (2011). DOI: 10.1155/2011/612353

## 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 ${F}_{n}(x)=F(x)\ast{\delta }_{n}(x)$ for $n=1,2,\dots$ and $\{{\delta }_{n}(x)\}$ is a certain regular sequence converging to the Dirac delta function. In the ordinary sense, the composition ${\delta }^{(s)}[({\mathrm{sinh}}^{-1}{x}_{+}{)}^{r}]$ does not exists. In this study, it is proved that the neutrix composition ${\delta }^{(s)}[({\mathrm{sinh}}^{-1}{x}_{+}{)}^{r}]$ exists and is given by ${\delta }^{(s)}[({\mathrm{sinh}}^{-1}{x}_{+})^{r}]=\sum_{k=0}^{sr+r-1}\sum_{i=0}^{k}\binom{\text{k}}{\text{i}}((-1)^{k}r{c}_{s,k,i}/{2}^{k+1}k!){\delta}^{(k)}(x)$, for $s=0,1,2,\dots$ and $r=1,2,\dots$, where ${c}_{s,k,i}=(-1)^{s}s![(k-2i+1)^{rs-1}+(k-2i-1)^{rs+r-1}]/(2(rs+r-1)!)$. Further results are also proved.

## Citation

Brian Fisher. Adem Kılıçman. "On the Neutrix Composition of the Delta and Inverse Hyperbolic Sine Functions." J. Appl. Math. 2011 1 - 12, 2011. https://doi.org/10.1155/2011/612353

## Information

Published: 2011
First available in Project Euclid: 12 August 2011

zbMATH: 1238.46030
MathSciNet: MR2800587
Digital Object Identifier: 10.1155/2011/612353