The author investigates solutions of the general homogeneous linear second order differential equation of the form $${{d^2w}\over{dz^2}}+ f(z){{dw}\over{dz}}+ g(z)w=0. \tag$*$ $$ In § 2, the author gives error bounds which are uniformly valid for $0\leq|\text{arg}(ze^{-\pi(j- 1)i})|\leq 2\pi$. In § 3, he gives details of the proof of the derivation of these bounds, which uses the technique of successive approximations. In § 4, the author generalizes the results of §§ 2-3 to give exponentially improved expansions with an improved relative error term of $O(z^{-m})$ as $z\to\infty$ where $m$ is a prescribed fixed positive integer. In § 5, he gives brief details on the extension of the error analysis to sectors, in conjunction with the results of § 4. In § 6, the author examines in more detail the asymptotic nature of the error bounds, which involve so-called weight functions. Finally, in § 7, he gives a numerical example on the calculation of certain constants which appear.