620-332 Integral Transforms & Asymptotics

Credit Points

Coordinator

Prerequisites

620-331, and one of 620-221, 620-252.

Semester

Contact

Subject Description

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

Assessment

Up to 24 pages of written assignments and a 3-hour end-of-semester written examination.