620-332 Integral Transforms and Asymptotics

Note

Students may only gain credit for one of 620-332, 618-332 (1997 Handbook), 618-331 (1996 Handbook).

Credit Points

Coordinator

Prerequisites

620-331 (1996 Handbook 618-332; 1997 Handbook 618-331), and one of 620-221, 620-252, 618-202 (1996 Handbook), 618-201 (1997 Handbook), 618-252 (1997 Handbook).

Semester

Contact

Subject Description

Students completing this subject should comprehend

• how to evaluate real integrals using complex analysis;
• how to evaluate and invert Fourier, Laplace and Mellin transforms, with selected applications including summing series and computing asymptotic series;
• 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;
• how to find asymptotic solutions to ordinary differential equations;

have developed the ability to

• apply a range of mathematical techniques to correctly solve applied mathematics problems;

and appreciate

• the power of these techniques to solve mathematical problems.

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 three-hour end-of-semester written examination.