On approximate solutions of linear differential equations

Asymptotic approximations of Green's type to solutions of differential equations are studied, with special reference to the uniformity of the approximation given by the first term. In extension to the complex variable this is found to require substantial restrictions on the region considered. An anomaly previously noticed is traced to non-uniformity of approximation. The case where the coefficient χ 0 has a simple zero and χ 1 is not zero is treated by a simple method.