Mathematical Sciences: Uniform Asymptotic Expansion of Integrals

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This research will consider several problems associated with the asymptotic expansions of functions defined by integrals. Realistic error bounds will be constructed for uniform asymptotic expansions of associated Legendre functions of large degree and fixed order directly from the integral representations of these functions. The classical method of Chester, Friedman, and Ursell will be generalized by mapping phase functions into rational functions and logarithms. The derivation of computable error bounds will be emphasized. This work has applications in the area of special functions. Such functions arise quite naturally in a variety of physical contexts in both fluid dynamics and solid mechanics.