620-332 Integral Transforms & Asymptotics

One of 620-232 or 620-234; and one of 620-221 or 620-252.

This subject introduces methods of evaluating real integrals using complex analysis; and develops methods for evaluating and inverting Fourier, Laplace and Mellin transforms, with selected applications including summing series and computing asymptotic series. Students should learn what an asymptotic expansion is and how it provides approximations; how to use Watson's lemma and the methods of Laplace, stationary phase and steepest descents to evaluate asymptotic expressions; and how to find asymptotic solutions to ordinary differential equations. This subject demonstrates a range of important and useful techniques and their power in solving problems in applied mathematics.

Complex analysis topics include contour integration, branch cuts, and evaluation of integrals. Integral transforms topics include wave equation, Fourier series; Fourier transform, Fourier integral theorem, convolution, applications; Laplace transform, inversion, examples; application to ordinary differential equations; convolution and applications; and Mellin transform examples. Asymptotics topics include asymptotic expansions, application of Mellin transform; Laplace's method for integrals, method of steepest descent, applications; method of stationary phase, examples; and WKB method for ordinary differential equations, asymptotic matching.

Up to 24 pages of written assignments, a 3-hour end-of-semester written examination and class tests totalling not more than 1.5 hours.