THE NEUTRIX PRODUCT OF THE DISTRIBUTIONS

Abstract

In this paper author has obtained the neutrix product of x+ −r and δ (α) (x), where α is a positive fractional number. Key Words: Neutrix limit, fractional-differentiation, sequence, function. Ams Subject Classification- 46F 1. INTRODUCTION Neutrix N is defined by J.G. vander Corput [2] as a commutative additive group of functions ν(ξ) defined on a domain N' with values in additive group N", where further if for some ν in N, ν(ξ) = γ for all ξ in N', then γ= 0. The functions in N are called negligible functions. Now let N' be a set contained in a topological space with a limit point b which does not belong to N'. If f(ξ) is a function defined on N' with values in N" and it is possible to find a constant β such that f(ξ) –β is negligible in N, then β is called the neutrix limit or N-limit of f as ξ tends to b and we write N− lim ξ→b f(ξ) = β, where β must be unique, if it exists. Introducing the neutrix limit, Fisher [3,4] defined the neutrix product of two distributions as –