Syllabus

Fourier analysis is a key mathematical tool in a scientists arsenal, and is used across the all fields in then natural sciences as well as other disciplines such as engineering. The first topic is that of the Fourier series, which allows for the representation of a periodic function as a summation of sinusoidal components, which has a wide variety of applications such as signal processing and solving partial differential equations like the heat diffusion equation. The second topic is that of integral transforms, specifically Fourier and Laplace transforms. The former is a natural extension of Fourier series, with an equally wide range of applications. For example; some time varying sound wave can be instead represented as a distribution function in frequency space rather than the usually spatial coordinates.

Details

This class would to take place once a week over a half a semester. We followed the textbook Fourier Series by Tolstov. The material was the subject of seminars held immediately after each lecture.

Class slides

PDFs of lecture slides can be found in the FA folder at this url. Note these slides were used in classes I taught 2021 and have not been updated since.

Part 1: Introduction, derivation of Fourier series

Part 2: Properties of Fourier series

Part 3: Applications of Fourier series

Part 4: Integral transforms

Part 5: Solving PDEs with Fourier analysis