Mathematica Eterna
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Abstract

On the Neutrix Composition of Distributions of the Delta Function and the Function [cosh−1 + (x + 1)]r

Fatma Al-Sirehy

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 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) for n = 1, 2, . . ., 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 proved that the neutrix composition δ (s) [cosh−1 + (x + 1)]r exists and δ (s) [cosh−1 + (x + 1)]r = rsX +r−2 k=0 X k j=0 X j i=0 (−1)s+k−j s! r2 j+2  k j j i  × [(j − 2i + 1)rs+r−1 − (j − 2i − 1)rs+r−1 ] (rs + r − 1)! δ (k) (x), for r, s = 1, 2, . . . . .