THE NEUTRIX PRODUCT OF THE DISTRIBUTIONS x_(+ )^(-r) AND δ^((α) ) (x)

THE NEUTRIX PRODUCT OF THE DISTRIBUTIONS

Sanjeev Srivastava¹, Chinta Mani Tiwari²

^1,2Department of Mathematics (School of Science)

Maharishi University of Information Technology, Lucknow, Uttar Pradesh (India)

---------------------------------------------------------------------***---------------------------------------------------------------------

Abstract – In this paper author has obtained the neutrix product of  where  is a positive fractional number.

Key Words:  Neutrix limit, fractional-differentiation, sequence, function.

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) =  for all ξ in N', then = 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  such that f(ξ) –  is negligible in N, then  is called the neutrix limit or N-limit of f as ξ tends to b and we write

where  must be unique, if it exists.