In this project the principal investigator will derive uniform asymptotic approximations for the solutions of second- order linear ordinary differential equations that contain a large parameter. The approximations would be uniformly valid in a domain of the complex plane that contains a coalescing turning point and a simple pole. In the beginning of the project the principal investigator will concentrate his efforts on approximating solutions of Mathieu's equation, including constructing realistic and explicit error bounds. He then hopes to extend his techniques and results to more general equations with turning points. Linear, second-order ordinary differential equations are in a real sense the building blocks for many complex theories in mathematical physics and asymptotic analysis. The solutions of such equations are useful in their own right as the solutions of interesting and important physical problems and equally as paradigms for more complicated phenomena. In this project the principal investigator will seek to find accurate approximate solutions of such an equation, known as Mathieu's equation, that arises in a number of applied contexts. Its solutions are known eponymously as Mathieu functions, and their properties are representative of asymptotic properties of solutions of more general equations involving turning-points.
