Asymptotic Analysis 1 (1988) 317-350 N orth-Holland 317 AN INVARIANT ASYMPTOTIC FORMULA FOR SOLUTIONS OF SECOND-ORDER LINEAR ODE'S * H. GINGOLD Department of Mathematics, West-Virginia University, Morgantown, WV 26506, US.A. Received July 1987 Revised March 1988 Abstract An invariant matrix asymptotic formula for the approximation of solutions of second-order linear ordinary differential equations y" = <p(x)y is proposed and elaborated upon. This is utilized to develop scalar versions of asymptotic formulas for two linearly independent solutions and for their derivatives. This formula is shown to be valid in a half neighbourhood of a point Xo. The validity holds whether Xo is an ordinary (regular) point for the ODE, whether Xo is a singular regular point for the ODE, if some exceptional case is avoided, whether Xo is a singular irregular point for the ODE, and whether or not Xo is finite. The matrix version of our formulas is shown to be valid also at a turning point. The Liouville-Green approximation is extracted as a particular case of our formula. Examples are given. The formula has additional" globality" properties. Examples are given where the ODE is considered on an infinite interval (0,00) and its coefficient <p(x) is singular at x = ° as well as at x = 00. A uniformly valid approximation on the entire infinite interval is then provided. 1. Towards invariant and global formulas There exists a voluminous literature in the mathematical sciences which develops asymptotic formulas for the approximation of solutions of a second-order linear differential equation y" = <p(x)y. (1.1) The Liouville-Green approximation sometimes also called "the WKB approximation" seems to be a most celebrated example. The attention that the second-order linear differential equation (1.1) attracted during the past two centuries is well deserved. This is so because of the frequent occurrence of this equation in problems of mathematical physics. The voluminous amount of corrections and generalizations to the L-G approximation, which was derived by scientists, and the vast additional asymptotic formulas developed by mathematicians (usually in connection with spectral theory and oscillation) indicate that an optimal state of affairs may have not been reached yet. We believe this statement to be true not only for coefficients <p(x) which belong to an exotic family of functions on the interval [a, b], but also for a family of functions <p(x) which "behave like meromorphic functions" or are meromorphic at x 0 on [a, b]. One comes across various asymptotic formulas which are valid as x 00 but are invalid if the singularity of the equation is located at a finite point and vice versa (xo could be a singularity of the equation * This research was supported in part by Grant NAG-1-741 from the National Aeronautics and Space Administra- tion.