Basic Results in Asymptotics

Abstract

In many problems of engineering and the physical sciences we attempt to write the solutions as infinite series of functions. The simplest series representation is the power series. Given a function f(x) of a real variable x containing a number x0 in its domain of definition, we try to find a power series of the form $$\sum^{\infty}_{j=0}{a_{j}(x-x_{0})}^{j}$$ (1.1.1.) which provides a valid representation of f(x) in the interval I of convergence of the power series. It emerges that if f(x) has uniformly bounded derivatives of all orders at each point in I, the above series is uniquely determined and $$a_{j} = \frac{f^{(j)}(x_{0})}{j!}$$ where f(j)(x0) is the j-th derivative of f(x) evaluated at x0- Then the series (1.1.1) is called the Taylor series.KeywordsAsymptotic ExpansionAsymptotic AnalysisBasic ResultConvergent SeriesAsymptotic SeriesThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.