Asymptotic Approximations Of Integrals PDF Download

This thesis analyzes asymptotic approximations expansions for integrals, with several examples given. The main part of this research consists in studying the function g(x) = (1 + 1=x)x, which has the limit e as x → ∞. In a 2014 paper C.-P. Chen and J. Choi previously studied this function through an asymptotic expansion for large x. Our main focus is on the coefficients that appear in their expansion. Chen and Choi obtained an explicit formula for the coefficients, but it involves a sum of terms that grows exponentially in number. Our contribution is to find the coefficients in a more practical way, and also to determine the asymptotic behavior of the nth term as n → ∞. We derive a recursion formula, and we show it is simple to use and is numerically stable. We then use Cauchy’s integral formula to derive an explicit integral representation for the coefficients. From this we approximate the late coefficients by residue theory, and this approximation consists of two simple terms. We show the accuracy of the approximation with some numerical examples. We finally determine an integral representation of the error term in our asymptotic approximation, and from this show that is of smaller order of magnitude than the two leading terms.