Generalized functions using the neutrix calculus.

In Chapter 1, we give a brief review of the basic properties of distributions and we also present neutrices and neutrix limits which are needed to define the product and convolution product of distributions. In Chapter 2, the product of two distributions f and g is defined to be the neutrix limit of the {lcub}fgn{rcub}, provided this limit exists, where gn = 9 n and is a regular sequence converging to the Dirac delta function. The neutrix product f o g is said to exist and be equal to h if [Mathematical equation removed] for all in D. Some theorems about the existence of this product for distributions are proved. The neutrix product of distributions in this chapter is in general non-commutative. In Chapter 3, we define the commutative neutrix product of distributions. Neutrix products of the form [Mathematical equation removed] are evaluated from which further neutrix products are obtained. In Chapter 4, we let f and g be distributions in D' and let fn{lcub}x) = f{lcub}x)Tn(x), where {lcub}Tn(x){rcub} is a certain sequence of functions which converges to the identity functions as n tends to infinity. The neutrix convolution product f g is then defined as the neutrix limit of the sequence {lcub}fn g{rcub}, provided the limit h exists in the sense that [Mathematical equation removed] for all in D. The neutrix convolution product is evaluated for [Mathematical equation removed] The convolution product of distributions in this chapter, is in general non-commutative. In Chapter 5, we consider a commutative neutrix convolution product f * g of distributions and evaluate various neutrix convolution products of distributions. Finally, in Chapter 6, we define a new neutrix product f g on the space of ultradistributions Z'. A new commutative neutrix convolution product f g of two distributions f and g in D' has recently been defined. If f = f (f) and g = f (g) are the Fourier transforms of f and g respectively, then the neutrix product f g is defined by the exchange formula [Mathematical equation removed].