Title: Chapter 6 Asymptotic Expansions – The Formal Concept
Abstract: Publisher SummaryThis chapter discusses the concept of an asymptotic sequence, and defines the notion of an asymptotic expansion of a function. It discusses that the asymptotic power series expansion of the integral of a function is obtained from the asymptotic power series expansion off by integrating term by term. This is based on the assumption that the improper integral off converges. The asymptotic power series expansion of the derivative of a function f (if the expansion exists) is obtained from the asymptotic power series expansion off by differentiating term by term. The theory of this chapter extends to functions of a complex variable. In the complex plane, z approaches ∞ along many paths; whereas in the real plane essentially just one path, the positive x-axis, is available. It turns out that an asymptotic expansion of an analytic function is usually not valid over all paths that approach ∞, but is valid if the paths are restricted to a certain sector of the complex plane, a sector which depends on the function involved. The results about asymptotic expansions of a function of a real variable carry over to functions of a complex variable with only minor qualifications. The uniqueness of the coefficients of an asymptotic expansion off, relative to a given asymptotic sequence, is valid only over a sector of the complex plane. An analytic function is different asymptotic expansions, each involving the same asymptotic sequence, over different sectors of the complex plane; this is known as Stokes' phenomenon.
Publication Year: 1975
Publication Date: 1975-01-01
Language: en
Type: book-chapter
Indexed In: ['crossref']
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