On the Composition and Neutrix Composition of the Delta Function with the Hyperbolic Tangent and Its Inverse 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. It is proved that the neutrix composition δ(rs−1)((tanhx+) 1/r) exists and δ(rs-1)((tanhx+) 1/r)=∑k=0s-1∑i=0Kk((-12) kcs-21i-,k(rs)!/sk!)δ(k)(x) for r, s = 1,2, …, where Kk is the integer part of (s − k − 1)/2 and the constants cj,k are defined by the expansion (tanh-1x) k={∑i=0∞(x21i+/(21i+))}k=∑j=k∞cj,kxj, for k = 0,1, 2, …. Further results are also proved.

Date: 2011
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https://doi.org/10.1155/2011/846736

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Persistent link: https://EconPapers.repec.org/RePEc:wly:jnljam:v:2011:y:2011:i:1:n:846736

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