Journal of Applied Mathematics
Volume 2011 (2011), Article ID 846736, 13 pages
http://dx.doi.org/10.1155/2011/846736
Research Article

On the Composition and Neutrix Composition of the Delta Function with the Hyperbolic Tangent and Its Inverse Functions

Brian Fisher1 and Adem Kılıçman2

1Department of Mathematics, University of Leicester, Leicester LE1 7RH, UK
2Department of Mathematics and Institute for Mathematical Research, University Putra Malaysia, Serdang, 43400 Selangor, Malaysia

Received 20 April 2011; Accepted 13 June 2011

Academic Editor: Andrew Pickering

Copyright © 2011 Brian Fisher and Adem Kılıçman. 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.

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 { 𝐹 𝑛 ( 𝑓 ( 𝑥 ) ) } is equal to ℎ ( 𝑥 ) , where 𝐹 𝑛 ( 𝑥 ) = 𝐹 ( 𝑥 ) ∗ 𝛿 𝑛 ( 𝑥 ) for 𝑛 = 1 , 2 , … and { 𝛿 𝑛 ( 𝑥 ) } is a certain regular sequence converging to the Dirac delta function. It is proved that the neutrix composition 𝛿 ( 𝑟 𝑠 − 1 ) ( ( t a n h 𝑥 + ) 1 / 𝑟 ) exists and 𝛿 ( 𝑟 𝑠 − 1 ) ( ( t a n h 𝑥 + ) 1 / 𝑟 ∑ ) = 𝑠 − 1 𝑘 = 0 ∑ 𝐾 𝑘 𝑖 = 0 ( ( − 1 ) 𝑘 𝑐 𝑠 − 2 𝑖 − 1 , 𝑘 ( 𝑟 𝑠 ) ! / 2 𝑠 𝑘 ! ) 𝛿 ( 𝑘 ) ( 𝑥 ) for 𝑟 , 𝑠 = 1 , 2 , … , where 𝐾 𝑘 is the integer part of ( 𝑠 − 𝑘 − 1 ) / 2 and the constants 𝑐 𝑗 , 𝑘 are defined by the expansion ( t a n h − 1 𝑥 ) 𝑘 ∑ = { ∞ 𝑖 = 0 ( 𝑥 2 𝑖 + 1 / ( 2 𝑖 + 1 ) ) } 𝑘 = ∑ ∞ 𝑗 = 𝑘 𝑐 𝑗 , 𝑘 𝑥 𝑗 , for 𝑘 = 0 , 1 , 2 , … . Further results are also proved.