On the method of matched asymptotic expansions

Abstract The prescription for forming composite series, and a modified form of that procedure, are used systematically and extensively to derive uniform asymptotic approximations to various given functions which cannot be approximated uniformly by expansions of classical (Poincaré) type. The formal method of matched expansions is then applied to a boundary-value problem for an ordinary differential equation with a turning point. It is proved with the help of the uniform approximations found earlier that in this problem a restricted form of the asymptotic matching principle is valid, even when it is applied to truncated inner and outer expansions which do not overlap to the order of the terms being matched.