Improved error bounds for second-order differential equations with two turning points

Recently [2] 1, [3] , I developed a uniform asymptotic theory of second-order linear differential equations with two coalescing simple turning points, and applied the results to the associated Legendre equation_ Subsequently, in the course of writing a paper on connection formulas for multiple turning points [4] it became clear how to effect some improvements in two of the four cases treated in [2]. The purpose of the present note is to describe these improvements. It will be assumed that the reader is familiar with the results presented in [2], and the same notation will be used except where indicated otherwise. The next section introduces new auxiliary functions for the solutions of the modified Weber equation. The new form of the general approximation theorem is stated and discussed in the third (and concluding) section. An application of the results to the approximation of Whittaker functions with both parameters large will be published in due course.