An introduction to neutrix composition of distributions and delta function

The composition of the distribution g(s) (x) and an infinitely differentiable function f (x) having a simple zero at the point x = x0 is defined by Gel’fand Shilov by the equation g(s) (f (x)). It is shown how this definition can be extended to functions f (x) which are not necessarily infinitely differentiable or not having simple zeros at the point x = x0, by defining g(s) (f (x)) as the limit or neutrix limit of the sequence {g(s)n (f(x))} where {gn (x)} is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function g(x). A number of examples are given

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

Let F be a distribution in D[superscript 1] 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[subscript n](f(x))} is equal to h(x), where F[subscript n](x)= F(x) ∗ δ[subscript n](x) for n = 1, 2, . . . and {δ[subscript n](x)} is a certain regular sequence converging to the Dirac delta function. In the ordinary sense, the composition δ([superscript s])[(sinh[superscript −1]x[subscript +])[superscript r] does not exists. In this study, it is proved that the neutrix composition δ([superscript s])[(sinh[superscript −1]x[subscript +])[superscript r] exists and is given by δ([superscript s])[(sinh[superscript −1]x[subscript +])[superscript r] = ∑[superscript sr+r-1, subscript k=0] ∑[superscript k, subscript i=0] ([superscript k, subscript i]) ((-1)[superscript k] rc[subscript s,k,i]/2[superscript k+1]k!)δ([superscript k])(x), for s = 0, 1, 2, . . . and r = 1, 2, . . ., where c[subscript s,k,i] = (−1)[superscript s]s![(k − 2i + 1)[superscript rs−1] + (k − 2i − 1)[superscript rs+r−1]/(2(rs + r − 1)!). Further results are also proved