On the neutrix composition of the delta and inverse hyperbolic sine functions (Q555006)

Summary: Let \(F\) be a distribution in \(\mathcal D'\) 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 \(F_n(x) = F(x) \ast \delta_n(x)\) for \(n = 1, 2, \dots\) and \(\{ \delta_n(x)\}\) is a certain regular sequence converging to the Dirac delta function. In the ordinary sense, the composition \(\delta^{(s)}[(\text{sinh}^{-1} x_+)^r]\) does not exist. In this study, it is proved that the neutrix composition \(\delta^{(s)}[(\text{sinh}^{-1} x_+)^r]\) exists and is given by \(\delta^{(s)} [(\text{sinh}^{-1} x_+)^r] = \sum^{sr+r-1}_{k=0} \sum^{k}_{i=0} \binom ki ((- 1)^k rc_{s,k,i}/2^{k+1} k!) \delta^{(k)} (x)\), for \(s = 0, 1, 2, \dots\) and \(r = 1, 2, \dots\), where \(c_{s,k,i} = (-1)^ss![(k - 2i + 1)^{rs-1} + (k - 2i - 1)^{rs+r-1}]/(2(rs + r - 1)!)\). Further results are also proved.