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

Brian Fisher and Adem Kılıçman

Journal of Applied Mathematics, 2011, vol. 2011, issue 1

Abstract: Let F be a distribution in 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 {Fn(f(x))} is equal to h(x), where Fn(x) = F(x)*δn(x) for n = 1,2, … and {δn(x)} is a certain regular sequence converging to the Dirac delta function. In the ordinary sense, the composition δ(s)[(sinh −1x+) r] does not exists. In this study, it is proved that the neutrix composition δ(s)[(sinh −1x+) r] exists and is given by δ(s)[(sinh -1x+) r]=∑k=0sr+r-1∑i=0k(ki)((-1) krcs,k,i/2k+1k!)δ(k)(x), for s = 0,1, 2, … and r = 1,2, …, where cs,k,i = (−1) ss! [(k − 2i + 1) rs−1 + (k − 2i − 1) rs+r−1]/(2(rs + r − 1)!). Further results are also proved.

Date: 2011
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https://doi.org/10.1155/2011/612353

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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnljam:v:2011:y:2011:i:1:n:612353

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