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

Let F be a distribution inD′ 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 ∑sr r−1 k 0 ∑k i 0 ( k i ) −1 rcs,k,i/2 1k! δ k x , for s 0, 1, 2, . . . and r 1, 2, . . ., where cs,k,i −1 s! k − 2i 1 rs−1 k − 2i − 1 rs r−1 / 2 rs r − 1 ! . Further results are also proved.