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Title: On the composition and neutrix composition of the delta function with the hyperbolic tangent and its inverse functions
Authors: Fisher, Brian
Kılıcman, Adem
First Published: 2011
Publisher: Hindawi Publishing Corporation
Citation: Journal of Applied Mathematics, Volume 2011, Article ID 846736
Abstract: Let F be a distribution in D[superscript 1] 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[subscript n](f(x))} is equal to h(x), where F[subscript n](x)= F(x) ∗ δ[subscript n](x) for n = 1, 2, . . . and {δ[subscript n](x)} is a certain regular sequence converging to the Dirac delta function. It is proved that the neutrix composition δ([superscript rs-1])((tanh x[subscript +])[superscript 1/r]) exists and δ([superscript rs-1])((tanh x[subscript +])[superscript 1/r]) = ∑[superscript s-1, subscript k=0] ∑[superscript K[subscript k], subscript i=0] ((-1)[superscript k]c[subscript s-2i-1,k] (rs)!/2sk!)δ([superscript k])(x) for r,s = 1, 2, . . ., where K[subscript k] is the integer part of (s-k-1)/2 and the constants c[subscript j,k] are defined by the expansion (tanh[superscript -1]x)superscript k = {∑[superscript ∞, subscript i=0] (x[superscript 2i+1]/(2i + 1))}[superscript k] = ∑[superscript ∞, subscript j=k] c[subscript j,k]x[superscript j], for k = 0,1,2,.... Further results also provided.
DOI Link: 10.1155/2011/846736
ISSN: 1110-757X
eISSN: 1687-0042
Version: Publisher Version
Status: Peer-reviewed
Type: Journal Article
Rights: Copyright © 2011 B. Fisher and A. Kılıc¸man. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Appears in Collections:Published Articles, Dept. of Mathematics

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