Calcul différentiel et intégral II

1. Sequences and Series of Functions

• Concepts of convergence for sequences and series of functions (pointwise, absolute, uniform, and normal convergence)
• Passage to the limit and properties of continuity and differentiability
• Power series
• Fourier series associated with a periodic function (Dirichlet’s, Bessel’s, and Parseval’s theorems)

2. Integration of Real-Valued Functions over an Arbitrary Interval

• Absolutely convergent integrals and convergent integrals (Abel’s criterion, square-integrable functions, Cauchy-Schwarz inequality)
• Parameter-dependent integrals
• Application to the Fourier transform in the Schwartz class (inversion and Plancherel theorems)

3. Differential Equations

• General concepts, the notion of a Cauchy problem (initial value problem), reduction to first order, integral formulation
• Local Cauchy-Lipschitz theorem (Picard-Lindelöf theorem)
• Maximal solutions and the global Cauchy-Lipschitz theorem
• Linear differential equations and systems (Gronwall’s lemma, Duhamel’s formula, matrix exponential in the case of constant coefficients)

4. Functions of a Complex Variable

• Complex differentiability and the Cauchy-Riemann equations
• Holomorphic functions, Cauchy’s theorem in a neighborhood and then in a convex domain, and representation as a power series
• Applications: principle of isolated zeros, classification of isolated singularities, Liouville’s theorem, the Fundamental Theorem of Algebra, and Cauchy estimates
• Homotopy of closed paths and the global Cauchy theorem
• Residue theorem.

This course constitutes the second part of the Differential and Integral Calculus sequence and introduces the fundamental concepts of mathematical analysis that are essential for the study of physics and advanced topics in mathematics.

The four most important concepts that students will have mastered by the end of the course are:

• Different notions of convergence in infinite-dimensional spaces of functions
• A generalized notion of integration (over intervals that are not necessarily closed and bounded)
• Differential equations, the phenomena they model, and their underlying structure
• Differentiability of functions of a complex variable.

The course includes numerous applications, with particular emphasis on frequency analysis through Fourier series and the Fourier transform, which will reappear throughout the course in a variety of contexts.

Required and corequired courses

Courses requiring this course

Cours ayant celui-ci comme co-requis

The course consists of lectures, exercise sessions, and personal project.

References, bibliography, and recommended reading

Course syllabus (including references to external literature).

Course notes

• Syllabus

Contacts

Mélanie Bertelson (Melanie.Bertelson@ulb.be) et Jean Gutt (Jean.Gutt@ulb.be)

Campus Plaine, Bâtiment NO, 7ème étage

Campus

Plaine

Method(s) of evaluation

• Personal work
• written examination

Personal work

written examination

Les examens partiels de janvier et de juin seront des écrits qui comporteront des questions de théorie ainsi que des exercices à résoudre. Deux travaux personnels seront proposés chaque semestre.

Mark calculation method (including weighting of intermediary marks)

The final grade is a number out of 20 and is calculated as follows. At the end of each semester, an examination is held covering the material taught during that semester. You receive a grade, to which a bonus is added which is equal to the grade you obtained for your personal work for that semester divided by 10.

The final grade is the arithmetic mean of the two semester grades, provided that each of them is at least 7/20. If this condition is not met, the final grade is capped at 7/20.

If, at the end of the first examination session, you have not validated the course, you will be required to retake in the second session any semester for which you did not obtain at least 10/20 (including the bonus points) during the first session. After the second-session examination(s), your final grade will be recalculated using the same method as in the first session, taking into account the most recent grades obtained. The bonus points earned during the academic year will also be added.

The following examples should help clarify the system:

• Bob obtains 8/20 on the January examination, with a 0.5-point bonus for Semester 1, and 7/20 on the June examination, with a 1-point bonus for Semester 2. His final grade is therefore (8+0.5+7+1)/2 which yields 8.5/20 (the nearest half integer to 8,25). If Bob wishes to pass the course during the same academic year, he must retake both semesters. If he then obtains 7/20 for Semester 1 and 12/20 for Semester 2, his final grade becomes (7+0.5+12+1)/2, which yields 10.5/20, and Bob passes the course.
• Alice obtains 14/20 for Semester 1 and 6/20 for Semester 2. Her final grade will be 7/20 (and not 10/20), because one of her semester grades is below 7/20. She must therefore retake Semester 2. If she obtains 5/20 for Semester 2 in the second session, her final grade will remain 7/20. If she improves her Semester 2 grade to 7/20, her final grade will become 10.5/20, and she will pass the course.

If anything remains unclear, please do not hesitate to ask your lecturers for further clarification.

Language(s) of evaluation

• french