Mastery of complex numbers and functions of complex variables. Study of the convergences of series with complex terms and functions of complex variables. Study of Fourier, Taylor and Laurent series. Application to the calculation of integrals via the residue theorem.
The aim of this course is to teach the theoretical and practical bases of complex analysis that a future physicist or mathematician must have. Its purpose is to study functions of complex variables and in particular series of functions (Taylor, Laurent and Fourier). This course is a logical extension of the Analysis course given in the first bachelor's degree and it pursues the same objectives as it.
The following subjects will be covered in the course:
Chapter 0: Complex numbers
Chapter 1: Integer series. Analytic functions of a complex variable.
Chapter 2: Elementary theory of Fourier series.
Chapter 3: Integral along a path
Chapter 4: Analytic functions
Chapter 5: Cauchy's theorem and integral formula and their consequences
Chapter 6: Singularities and Laurent series
Chapter 7: Cauchy's residue theorem and applications to the calculation of integrals on a contour
The exam consists of two parts: a written exam (exercises) and an oral exam (course questions).
1) The written test: 3 hours with exercises taken from those seen in class/tutorials and in the syllabus
2) the oral test: presentation on the board of a course question in 10-15 minutes, extracted in advance, without notes.
The exam grade is calculated as follows:
- if the student takes both tests and obtains at least 2/20 in each test, then we calculate N1 = (2 x Written grade + 1 x Oral grade)/3. If this N1 grade is greater than or equal to 8/20 then we add the points of the TG/5.
- if the student takes both tests and obtains less than 2/20 in at least one test or signs one of the two tests, then the total mark will be 0 (SG)
If the total mark is strictly less than 10/20, then the student can carry over from one examination session to the next (in the same academic year) the written-oral marks if they are higher than 5/20.
The mark for the group work is automatically carried over from one examination session to the next (in the same academic year).
Syllabus for the theoretical course for the exercises containing the reminders and the statements of the exercises per chapter. Reference book: H.A. PRIESTLEY, Introduction to complex analysis, Oxford Sciences Publications, 1990.
| Training | Study programme | Block | Credits | Mandatory |
|---|---|---|---|---|
| Bachelier en sciences mathématiques | Standard | 0 | 5 | |
| Bachelier en sciences informatiques | Standard | 0 | 5 | |
| Bachelier en sciences informatiques | Standard | 2 | 5 | |
| Bachelier en sciences mathématiques | Standard | 2 | 5 |