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.
D](s) maybe improved by including more terms of Stirling's asymptotic series .
where the limit of the integral on the rhs is a regular function for u > 5/2 and the asymptotic series A(w) converges for u > 1 /2.
Where the symbol indicates that the right-hand side is a so-called asymptotic series
for the left-hand side.
In order to circumvent the problem of the divergence of the asymptotic series, several possibilities have been presented in the literature.
The purpose of this paper is to analyze some features of this modification, namely the convergence properties of the modified asymptotic series and some techniques that can be used to compute the confluent hypergeometric functions involved in the approximation.
Birkhoff, Quantum mechanics and asymptotic series
Lemma 6 For s > 0 we have the asymptotic series
In this case the integral in (4) has typically an asymptotic series
expansion which is a linear combination of terms of order ([[parallel]([n.
SIMON, Meromorphic Szegjfunctions and asymptotic series
for Verblunsky coefficients, Acta Math.
The topic of this book for graduate students and researchers is the study of singular perturbations of ordinary differential equations, that is, perturbations that represent solutions as asymptotic series
rather than as analytic functions in a perturbation parameter.