Basic Results in Asymptotics

Ricardo Estrada and Ram P. Kanwal
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Ricardo Estrada: Universidad de Costa Rica, Escuela de Matemática
Ram P. Kanwal: The Pennsylvania State University, Department of Mathematics

Chapter Chapter 1 in Asymptotic Analysis, 1994, pp 1-42 from Springer

Abstract: 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 1.1.1. $$\sum^{\infty}_{j=0}{a_{j}(x-x_{0})}^{j}$$ 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.

Keywords: Asymptotic Expansion; Asymptotic Analysis; Basic Result; Convergent Series; Asymptotic Series (search for similar items in EconPapers)
Date: 1994
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DOI: 10.1007/978-1-4684-0029-8_1

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