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Title: On the Neutrix Composition of the Delta and Inverse Hyperbolic Sine Functions
Authors: Fisher, Brian
Kılıcman, Adem
First Published: 2011
Publisher: Hindawi Publishing Corporation
Citation: Journal of Applied Mathematics, Volume 2011, Article ID 612353
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. In the ordinary sense, the composition δ([superscript s])[(sinh[superscript −1]x[subscript +])[superscript r] does not exists. In this study, it is proved that the neutrix composition δ([superscript s])[(sinh[superscript −1]x[subscript +])[superscript r] exists and is given by δ([superscript s])[(sinh[superscript −1]x[subscript +])[superscript r] = ∑[superscript sr+r-1, subscript k=0] ∑[superscript k, subscript i=0] ([superscript k, subscript i]) ((-1)[superscript k] rc[subscript s,k,i]/2[superscript k+1]k!)δ([superscript k])(x), for s = 0, 1, 2, . . . and r = 1, 2, . . ., where c[subscript s,k,i] = (−1)[superscript s]s![(k − 2i + 1)[superscript rs−1] + (k − 2i − 1)[superscript rs+r−1]/(2(rs + r − 1)!). Further results are also proved.
DOI Link: 10.1155/2011/612353
ISSN: 1110-757X
eISSN: 1687-0042
Version: Publisher Version
Status: Peer-reviewed
Type: Journal Article
Rights: Copyright © 2011 B. Fisher and A. Kılıcman. 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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