Error bounds for the Liouville–Green (or WKB) approximation

Error bounds are derived and examined for approximate solutions in terms of elementary functions of the differential equations in which u is a positive parameter, the functions f and p are free from singularities and p does not vanish. Bounds are also obtained for the remainder terms in the asymptotic expansions of the solutions in descending powers of u . The variable x ranges over a real interval, finite or infinite or over a region of the complex plane, bounded or unbounded. Applications are made to parabolic cylinder functions of large orders, and modified Bessel functions of large orders.