RESULTS ON THE COMPOSITION AND NEUTRIX COMPOSITION OF THE DELTA FUNCTION

The neutrix composition F(f(x))) of a distribution F(x) and a locally summable function f(x) is said to exist and be equal to the distribution h(x) if the neutrix limit of the sequence {Fn(f(x))} is equal to h(x), where Fn(x) = F(x) * δn(x) and {δn(x)} is a certain sequence of infinitely differentiable functions converging to the Dirac delta-function δ(x): It is proved that the neutrix composition δ(s){[exp+(x) - 1]r} exists and δ^(s) {[exp+ (x) − 1]^r } exists δ^(s) {[exp+ (x) − 1]^r }= ∑rs+r-1,k=0 ((-1)^(s+k) s!crs+r-1,k)/(2rk!) δ^(k) (x) for r = 1, 2,... and s =0,1, 2,... Further results are also proved.