On the Neutrix Composition of the Delta and Inverse Hyperbolic Sine Functions

Abstract

LetFbe a distribution inD'and letfbe a locally summable function. The compositionF(f(x))ofFandfis said to exist and be equal to the distributionh(x)if the limit of the sequence{Fn(f(x))}is equal toh(x), whereFn(x)=F(x)*δn(x)forn=1,2,…and{δn(x)}is a certain regular sequence converging to the Dirac delta function. In the ordinary sense, the compositionδ(s)[(sinh⁡-1x+)r]does not exists. In this study, it is proved that the neutrix compositionδ(s)[(sinh⁡-1x+)r]exists and is given byδ(s)[(sinh⁡-1x+)r]=∑k=0sr+r-1∑i=0k(ki)((-1)krcs,k,i/2k+1k!)δ(k)(x), fors=0,1,2,…andr=1,2,…, wherecs,k,i=(-1)ss![(k-2i+1)rs-1+(k-2i-1)rs+r-1]/(2(rs+r-1)!). Further results are also proved.