This work involves the construction of error-bound theories for a number of important and long-standing problems in asymptotic analysis. Attempts will be made to derive realistic error bounds for uniform asymptotic expansions of definite integrals with coalescing saddle points, and for uniform asymptotic expansions of differerential equations with two coalescing turning points in the complex plane. The role of exponentially small terms in asymptotic expansions will be examined. The possibility of simplifying some existing expansions so they may be more readily computed in the complex plane will also be considered. This work should be of great value to researchers in the physical, biological, and engineering sciences where many of the model equations, with proper scaling, involve large or small parameters.
