On Distributional Derivatives with Laurent Series Approach to its Products
Eziokwu, C. Emmanuel *
Department of Mathematic, College of Physical and Applied Science, Michael Okpara University of Agriculture, Umudike, Abia State, Nigeria.
*Author to whom correspondence should be addressed.
Abstract
The distributional (weak) derivatives are weak methods that generalize the idea of usual derivatives.
This was developed by the French mathematician Laurent Schwartz around 1950.
This work provides enough basic theoretical background to distributions and in addition aims at supplying the much desired definitions and results in distributions in particular for products and convolution of distributions, since they are in maximum demand for areas like quantum field theory.
To make this work meaningful, the basics of Laurent’s series was explored leading to the introduction of the Laurent’s series representation of \(x_+^{\lambda}\) and \(r^{\lambda}\) which was used to generate or derive the products \(x_+^{-k}\) . \(\delta ^ {(p)}\) (x) of one variable and r2m-n . \(\delta ^ {(2s)}\) (r)of n variable and finally we implied that
\(x_+^{-m}\) . \(x_+^{-l}\) = \(x_+^{-m-l}\) [7].
Keywords: Breeding for proteins, Distributions regularization, Cereal and legume proteins, partition of unity, Seed proteins, distributional support, pseudo-topologies laurent’s series, neutrix products