## Journal of Applied Mathematics

### 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 $\mathcal{D}'$ 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 $n=1,2,\dots$ and $\{{\delta }_{n}(x)\}$ is a certain regular sequence converging to the Dirac delta function. It is proved that the neutrix composition ${\delta }^{(rs-1)}((\mathrm{tanh}{x}_{+}{)}^{1/r})$ exists and ${\delta}^{(rs-1)}((\tanh x_{+})^{1/r})={\sum}_{k=0}^{s-1}{\sum}_{i=0}^{{K}_{k}}((-1)^{k}{c}_{s-2i-1,k}(rs)!/2sk!){\delta}^{(k)}(x)$ for $r,s=1,2,\dots$, where ${K}_{k}$ is the integer part of $(s-k-1)/2$ and the constants ${c}_{j,k}$ are defined by the expansion $(\tanh^{-1}x)^{k}={\{\sum_{i=0}^{\infty}(x^{2i+1}/(2i+1))\}}^{k}=\sum_{j=k}^{\infty}{c}_{j,k}{x}^{j}$, for $k=0,1,2,\dots$. Further results are also proved.

#### Article information

Source
J. Appl. Math., Volume 2011 (2011), Article ID 846736, 13 pages.

Dates
First available in Project Euclid: 15 March 2012

https://projecteuclid.org/euclid.jam/1331818616

Digital Object Identifier
doi:10.1155/2011/846736

Mathematical Reviews number (MathSciNet)
MR2820079

Zentralblatt MATH identifier
1236.46035

#### Citation

Fisher, Brian; Kılıçman, Adem. On the Composition and Neutrix Composition of the Delta Function with the Hyperbolic Tangent and Its Inverse Functions. J. Appl. Math. 2011 (2011), Article ID 846736, 13 pages. doi:10.1155/2011/846736. https://projecteuclid.org/euclid.jam/1331818616