Generalized Functions Using Neutrix Calculus

Generalized functions are the extension of classical functions and allow us to generalize the concept of derivative to all continuous functions. The generalized functions have many applications in various fields especially in science and engineering where the non-continuous phenomena that naturally lead to differential equations whose solutions are not ordinary functions, such as the Dirac-delta function therefore generalized functions can help us to develop an operational calculus in order to investigate linear ordinary differential equations as well as partial differential equations with constant and non-constant coefficients through their fundamental solutions. In the present work we extend certain operations such as multiplications and convolutions to the generalized functions by using the neutrix limit approach. The essential use of the neutrix limit is to extract the finite part from a divergent quantity as one has usually done to subtract the divergent terms via rather complicated procedures in the renormalization theory. In fact we can consider the neutrices as the generalization of the Hadamard finite parts