ABSTRACT
This chapter discusses asymptotic expansion and perturbation analysis. The method of asymptotic expansion has been studied since the time of Euler, and, in fact, it is one of major weapons of Euler. The term “asymptotic series” was coined by Poincare whereas it was called “semi-convergent series” by Stieltjes and “convergently beginning series” by Emde. A number of studies considered transforming asymptotic expansions into convergent series, including Airey, van der CoIput, Miller, van Wijngaarden and Watson. The perturbation theory is related to the three-body problem, which considered the perturbations of the motion of two bodies being influenced by the existence of a much smaller third body. The method of perturbation is particularly useful in obtaining accurate approximate solutions for nonlinear differential equations. Such approximate solutions provide insight into the physics and mechanics of the problem, which computer generated simulations cannot produce.
