A note on the non-commutative neutrix product of distributions

Abstract

The distributionF(x ₊, −r) Inx₊ andF(x ₋, −s) corresponding to the functionsx ₊ ^−r lnx₊ andx ₋ ^−s respectively are defined by the equations

$$\left\langle {F(x_ + , - r)\ln x_ + ,\phi (x)} \right\rangle = \int_0^\infty {x^{ - r} \ln x\left[ {\phi (x) - \sum\limits_{i = 0}^{r - 2} {\frac{{\phi ^{(i)} (0)}}{{i!}}x^i \frac{{\phi ^{(i)} (0)}}{{(r - 1)!}}H(1 - x)x^{r - 1} } } \right]dx} $$

(1) and

$$\left\langle {F(x_ + , - s),\phi (x)} \right\rangle = \int_0^\infty {x^{ - s} \left[ {\phi (x) - \sum\limits_{i = 0}^{s - 2} {\frac{{\phi ^{(i)} (0)}}{{i!}}( - x^i )\frac{{\phi ^{(s - 1)} (0)}}{{(s - 1)!}}H(1 - x)x^{s - 1} } } \right]dx} $$

(2) whereH(x) denotes the Heaviside function. In this paper, using the concept of the neutrix limit due to J G van der Corput [1], we evaluate the non-commutative neutrix product of distributionsF(x ₊, −r) lnx₊ andF(x ₋, −s). The formulae for the neutrix productsF(x ₊, −r) lnx ₊ ox ₋ ^−s, x₊ ^−r lnx₊ ox ₋ ^−s andx ₋ ^−s o F(x₊, −r) lnx₊ are also given forr, s = 1, 2, ...