On the Neutrix Composition of the Delta and Inverse Hyperbolic Sine Functions - Brian Fisher; Adem Kılıçman (Journal of Applied Mathematics)
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 {Fn(f(x))} is equal to h(x), where Fn(x)=F(x)*δn(x) for n=1,2,… and {δn(x)} is a certain regular sequence converging to the Dirac delta function. In the ordinary sense, the composition δ(s)[(sinh⁡-1x+)r] does not exists. In this study, it is proved that the neutrix composition δ(s)[(sinh⁡-1x+)r] exists and is given by δ(s)[(sinh⁡-1x+)r]=∑k=0sr+r-1∑i=0k(ki)((-1)krcs,k,i/2k+1k!)δ(k)(x), for s=0,1,2,… and r=1,2,…, where cs,k,i=(-1)ss![(k-2i+1)rs-1+(k-2i-1)rs+r-1]/(2(rs+r-1)!). Further results are also proved.