Notions of topology, differential calculus (differential of a function, partial derivative)

The first part of the course is an introduction that is designed to expose the theory of complex analysis in a short amount of time. The attempts to solve the pain points experienced by attending students over the years explain most of the deviations from standard expositions. We will skip some technical proofs (that will still be available for the most motivated students), focusing on the meaning of the different concepts. The second part of the course focuses on use cases of complex analysis in engineering. Numerous modern problems in many unrelated fields (scientific computing, image editing, signal processing, control problems) can be solved by complex analysis even though their statements have nothing to do with this theory.

During the first part of the course (~3 days), we will alternate 1h20 lectures with 1h20 exercise sessions.

During the second part of the course, we will focus on several applications. They will be presented as guided problems (half a day for each application), and the students will work together in small groups. The professor will regularly correct the different steps. Alternatively, extended theoretical exercise sessions could also be offered to volunteers.

Provisional schedule:

Day 1: General introduction, complex-differentiability (Cauchy-Riemann equations), holomorphic functions, line integrals, primitives, concepts of topology

Day 2: topology (path-connected/ connected sets, set operations, components), winding number, Cauchy's integral theorem, and corollaries.

Day 3: Power series (Laurent series), Zeros & Poles, isolated singularities, computation of residues, application to integrals of functions of a real variable

Day 4: Morning-Application to image editing Afternoon- Application to control theory (Laplace transform, stability criteria).

Day 5: Morning-Application to signal processing. Afternoon-Exam + debriefing.

Three short quizzes will be proposed during the week. They will count for 50% of the final mark. The questions will be straightforward applications of the definitions seen during the class. A final exam will take place on the last afternoon. It will count for 50% of the final mark.