  Recently, the present authors derived new asymptotic expansions for linear
differential equations having a simple turning point. These involve Airy
functions and slowly varying coefficient functions, and were simpler than
previous approximations, in particular being computable to a high degree of
accuracy. Here we present explicit error bounds for these expansions which only
involve elementary functions, and thereby provide a simplification of the
bounds associated with the classical expansions of F. W. J. Olver.
