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Further results on the neutrix composition of distributions involving the delta function and the function cosh-1+(x1/r + 1)

Fisher Brian, Tas Kenan

Demonstratio Mathematica

2019

Vol. 52, nr 1

249--255

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).The function cosh-1+(x + 1) is defined by cosh-1+(x + 1) =H(x) cosh-1(|x| + 1), where H(x) denotes Heaviside’s function. It is then proved that the neutrix composition δ(s)[cosh-1+(x1/r+1)] exists and (…), for r,s = 1,2,.... Further results are also proved. Our results improve, extend and generalize the main theorem of [Fisher B., Al-Sirehy F., Some results on the neutrix composition of distributions involving the delta function and the function cosh-1+(x + 1), Appl. Math. Sci. (Ruse), 2014, 8(153), 7629–7640].