Research Information System
Demonstratio Mathematica, vol.52, no.1, pp.249-255, 2019 (Scopus)
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 (x). The function cosh +-1(x+1) 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) ] delta (s)[cosh+ -1(x1/r + 1 )] exists and δ(s)[ cosh +-1(x1/r+1) ]=k=0s-1j=0kr+r-1i=0j(-1)kr+r+s-j-1r2j+2(kr+r-1j)(ji)[ (j-2i+1)s-(i-2i-1)s ]δ(k)(x), &delta (s)[cosh+ -1(x1/r + 1) ] = ∑limitsk = 0s-1 ∑j = 0kr + r - 1 ∑i = 0j (- 1)kr + r + s - j - 1r 2j + 2( + r - 1 cr j cr ) left(\cr i)[(j - 2i + 1)s-(i - 2i - 1 )s]δ(k)(x), 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].