Learning outcomes

Mastery of complex numbers and functions of complex variables. Study of Fourier, Taylor and Laurent series. Application to the calcolus of integrals.

Goals

Like the course of the 1st year of the bachelor's degree, this course aims to teach the theoretical and practical bases of Analysis that a future physicist or a future mathematician must possess. Its purpose is to study 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 it pursues the same objectives as the latter.

Content

The following topics will be covered in the course:
  1.      Chapter 0: Complex numbers
  2.      Chapter 1: Entire series. Analytic 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

 

Table of contents

  1.      Chapter 0: Complex numbers
  2.      Chapter 1: Entire series. Analytic 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

 

 

Assessment method

The exam has two parts: a written exam (exercises) and an oral exam (course questions).
 
1) The written test: 3 hours of exercises taken from those seen in class/TD and in the syllabus
 
2) the oral test: presentation on the blackboard of a lesson question in 10-15 minutes, extracted in advance, without notes.
 
The exam score is calculated as follows:
 
- if the student takes both tests and obtains at least 2/20 in each test, then N1 is calculated as (2 x written score + 1 x oral score)/3. If this note N1 is greater than or equal to 8/20 then the points of the TG/5 are added.
 
- if the student takes both tests and obtains less than 2/20 in at least one test or she signs one of the two tests, then the total mark will be 0 (SG)
 
If the total mark is strictly lower than 10/20, then the student can postpone from one exam session to the next (in the same academic year) the written or oral marks if they are higher than 5/20.
 
The group work grade is automatically carried over from one exam session to the next (in the same academic year).

 

 

Sources, references and any support material

Syllabus pour le cours théorique  pour les exercices contenant les rappels et les énoncés des exercices par chapitre.

Ouvrage de référence : H.A. PRIESTLEY, Introduction to complex analysis, Oxford Sciences Publications, 1990.

 

Language of instruction

French
Training Study programme Block Credits Mandatory
Bachelier en sciences physiques Standard 0 5
Bachelier en sciences physiques Standard 2 5