Asymptotic Expansions: Their Derivation and Interpretation - ISBN:0122165500

Asymptotic Expansions: Their Derivation and Interpretation

Date: 11 June 2011, 02:02

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Throughout this book, the designation "asymptotic series" will be reserved for those series in which for large values of the variable at all phases the terms first progressively decrease in magnitude, then reach a minimum and thereafter increase. The theory of such ultimately divergent series will be called the discipline of "asymptotics". (Frequently, in both mathematics and physics, these phrases have been allowed wider meanings; see footnote to Section 5 of Chapter I). Some examples of such series were discovered in the early eighteenth century by James Stirling, Leonard Euler and Colin Maclaurin, but for over a hundred years were rarely regarded seriously by pure mathematicians. The pure mathematicians of the early nineteenth century, and above all Augustin-Louis Cauchy and Neils Abel, placed too great a stress on convergence to trust such runaway expansions. However, notwithstanding the continued absence of a thorough general investigation, a number of important asymptotic expansions were soon being found.

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