Generalized functions

Abstract

Asymptotic analysis is an old subject that has applications in various fields of pure and applied mathematics, physics, and engineering. Asymptotic techniques are used to approximate complicated integral expressions resulting from various analytical transformation. Also, the solutions of some differential equations are calculated with greater precision using asymptotic techniques. An important feature of the theory of asymptotic expansions is that it plays an important role with experience and intuition in solving particular problems linked to nature. The objective of this study is to present approximations of this theory using generalized functions. The theory of generalized functions is another important area within mathematics, in which applications have been found for physics and engineering. Various studies have found a relationship between asymptotic techniques and distribution theory. The research is aimed at knowing in which distribution spaces such as E^,(R),P^,(R),O_γ^,(R).The expansion is valid. The momentum technique of asymptotic expansion is valid for many varieties of nuclei, such as fast-decaying and rapidly oscillating generalized functions. The asymptotic expansion also provides expansions of many integrals and series. Some series can be both convergent and asymptotic; however, there are divergent series that are asymptotic, which is why asymptotic series are referred to as divergent series. Rapidly decaying distributions at infinity are of fundamental importance in the study of generalized asymptotic expansions of functions. These expansions immediately give rise to the classical development of various integrals and series. The moment of the asymptotic expansion is referred to the expansion of the distributional kernels of the type f(λx) as λ→∞.