Mastery of complex numbers and functions of complex variables. Study of Fourier, Taylor and Laurent series. Application to the calculation of integrals.
Like the 1st year undergraduate course, this course aims to teach the theoretical and practical foundations of Analysis that a future physicist or mathematician should possess. It is aimed at studying questions more specifically dealt with in complex variables, including the theory of Fourier series. This course is a logical extension of the Analysis course given in the first bachelor's degree and pursues the same objectives as the latter.
The following subjects will be covered in the course: 1. Chapter 0: Complex Numbers 2. Chapter 1: Integer series. Analytical functions of a complex variable. 3. Chapter 2: Elementary theory of Fourier series. 4. Chapter 3: Prelude to Cauchy's Theorem (Integral along a path) 5. Chapter 4: Cauchy's Theorem 6. Chapter 5: Consequences of Cauchy's Theorem 7. Chapter 6: Singularities and Laurent Series 8. Chapter 7: Cauchy Residue Theorem
The examination consists of two parts: a written examination (exercises) and an oral examination (course questions). 1) The written test: 3 hours of exercises taken from those seen in the course/DD and in the syllabus 2) Oral test: presentation on the board of a course question in 10-15 minutes, extracted in advance, without notes. The exam grade is calculated in this way: - If the student takes both papers and obtains at least 2/20 in each paper, then N1 = (2 x Written mark + 1 x Oral mark)/3 is calculated. If this N1 score is greater than or equal to 8/20 then the TG/5 points are added. - if the student takes both papers and obtains less than 2/20 in at least one paper or signs off on one of the two papers, then the total mark will be 0 (SG) If the total mark is strictly less than 10/20, then the student may carry over from one examination session to the next (in the same academic year) the written and oral marks if they are higher than 5/20. The group work mark 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 mathématiques | Standard | 2 | 5 |