Course Overview

This course teaches fundamental concepts of complex analysis and some
of its classical applications.

Learning Achievement

Competence

Course prerequisites

Metric spaces, integration theory, power series

Grading Philosophy

> Intermediate assessment (mark I out of 20, duration 1h30 to 2h40
during the semester) and final exam (mark E out of 20, duration 3h in
December)

- final mark I/3+2E/3

> Rules and protocol for failures/re-sits is clearly described: in
case of failure, student have access to a second exam(mark S)

- the final mark is then max(I/3+2S/3,S)

Course schedule

Course content includes: 

> Holomorphic functions

- C-differentiability
- Cauchy-Riemann equations link with harmonic functions
- Complex power series.

> Cauchy Formula

- Integral along a regular curve, index
- Cauchy's integral

> Applications of Cauchy's integral

- Maximum Principal
- Liouville's Theorem
- Isolated zeroes
- Removable singularities
- Open mapping
- Morera Theorem
- argument

> Sequences and integrals of holomorphic functions

- Uniform convergence of sequences of holomorphic functions
- Integral of a family of holomorphic functions
- Complex logarithm
- Riemann Mapping Theorem

> Locale theory

- Singularities, Laurent series
- Residues
- Applications to the explicit computation of integrals, Fourier
transforms

Course type

Lectures

Online Course Requirement

Instructor

Other information

Duration: 14 weeks

Language of instruction: English
Mode of delivery: Face-to-face teaching

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