It provides a detailed construction of the asymptotic series
as well as the error bounds.
The topics include asymptotic series
and approximations, some nonstandard perturbation procedures, summing asymptotic series
, mating limiting asymptotic expansions, averaging homogenization, natural and forced vibrations of beams and plates, the stability of beams and plates, and some related problems.
This is formulated with the expansion of the electric field amplitude in an asymptotic series
over the parameter k, followed by the insertion of the series in (4), the classification of the terms of different order in separate equations and the solution of these equations .
Among the methods of asymptotic analysis that he explains are divergent asymptotic series
, phase-integral methods, asymptotic evaluation of integral, and boundary layer analysis.
He discusses the role of expansions and asymptotics in statistics, the basic properties of power series and asymptotic series
, the study of rational approximations to functions, and various applications, such as the use of the delta method for bias reduction, variance stabilization, and the construction of normalizing transformations, with a focus on asymptotic normality and efficiency of standard estimators.
See Hardy (1949, Theorem 136) for a related general theorem on generalized Borel summability of asymptotic series
D](s) maybe improved by including more terms of Stirling's asymptotic series
In order to circumvent the problem of the divergence of the asymptotic series
, several possibilities have been presented in the literature.
Birkhoff, Quantum mechanics and asymptotic series
Lemma 6 For s > 0 we have the asymptotic series
1981] describes a code using a mixture of power series, continued fractions, and asymptotic series
, but giving only a relative accuracy of [10.
94 here, we deal with asymptotic series
, not necessarily convergent ones.