On the Noncommutative Neutrix Product of Distributions

Let f and g be distributions and let g n = ( g * δ n ) ( x ) , where δ n ( x ) is a certain sequence converging to the Dirac-delta function δ ( x ) . The noncommutative neutrix product f ∘ g of f and g is defined to be the neutrix limit of the sequence { f g n } , provided the limit h exists in the sense that N‐ lim n → ∞ 〈 f ( x ) g n ( x ) , φ ( x ) 〉 = 〈 h ( x ) , φ ( x ) 〉 , for all test functions in 𝒟 . In this paper, using the concept of the neutrix limit due to van der Corput (1960), the noncommutative neutrix products x + r ln x + ∘ x − − r − 1 ln x − and x − − r − 1 ln x − ∘ x + r ln x + are proved to exist and are evaluated for r = 1 , 2 , … . It is consequently seen that these two products are in fact equal.