Journal of Applied Mathematics
Volume 2011 (2011), Article ID 612353, 12 pages
http://dx.doi.org/10.1155/2011/612353
Research Article

On the Neutrix Composition of the Delta and Inverse Hyperbolic Sine 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, 43400 UPM, Serdang, Selangor, Malaysia

Received 24 December 2010; Accepted 8 April 2011

Academic Editor: A. A. Soliman

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. In the ordinary sense, the composition 𝛿 ( 𝑠 ) [ ( s i n h − 1 𝑥 + ) 𝑟 ] does not exists. In this study, it is proved that the neutrix composition 𝛿 ( 𝑠 ) [ ( s i n h − 1 𝑥 + ) 𝑟 ] exists and is given by 𝛿 ( 𝑠 ) [ ( s i n h − 1 𝑥 + ) 𝑟 ∑ ] = 𝑠 𝑟 + 𝑟 − 1 𝑘 = 0 ∑ 𝑘 𝑖 = 0 k i ( ( − 1 ) 𝑘 𝑟 𝑐 𝑠 , 𝑘 , 𝑖 / 2 𝑘 + 1 𝑘 ! ) 𝛿 ( 𝑘 ) ( 𝑥 ) , for 𝑠 = 0 , 1 , 2 , … and 𝑟 = 1 , 2 , … , where 𝑐 𝑠 , 𝑘 , 𝑖 = ( − 1 ) 𝑠 𝑠 ! [ ( 𝑘 − 2 𝑖 + 1 ) 𝑟 𝑠 − 1 + ( 𝑘 − 2 𝑖 − 1 ) 𝑟 𝑠 + 𝑟 − 1 ] / ( 2 ( 𝑟 𝑠 + 𝑟 − 1 ) ! ) . Further results are also proved.