Please use this identifier to cite or link to this item: `http://hdl.handle.net/2381/28477`
 Title: On the composition and neutrix composition of the delta function with the hyperbolic tangent and its inverse functions Authors: Fisher, BrianKılıcman, 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 Links: http://www.hindawi.com/journals/jam/2011/846736/http://hdl.handle.net/2381/28477 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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