The noncommutative neutrix product of $x_{-}^{-s}\ln^mx_{-}$ and $x_{+}^{r}$

In distribution theory, the product of two distributions can only be defined under certain conditions. In this paper, by using Fisher's definition, it is proved that if Gm,s(x) denotes the distribution ( ln m+1x-)(s), then the noncommutative neutrix product [Formula: see text] exists and [Formula: see text] for m, s ≥ 1, where [Formula: see text]. Related neutrix products are then deduced.

On the Noncommutative Neutrix Product of Distributions

Letfandgbe distributions and letgn=(g*δn)(x), whereδn(x)is a certain sequence converging to the Dirac-delta functionδ(x). The noncommutative neutrix productf∘goffandgis defined to be the neutrix limit of the sequence{fgn}, provided the limithexists in the sense thatN‐limn→∞〈f(x)gn(x),φ(x)〉=〈h(x),φ(x)〉, for all test functions in𝒟. In this paper, using the concept of the neutrix limit due to van der Corput (1960), the noncommutative neutrix productsx+rlnx+∘x−−r−1lnx−andx−−r−1lnx−∘x+rlnx+are proved to exist and are evaluated forr=1,2,…. It is consequently seen that these two products are in fact equal.