Calcul différentiel et intégral II

1. Function spaces and convergences

• Convergence of sequences and series of functions (simple, absolute, uniform, normal)

• Derivability of sums of series of functions

• Converging and absolutely converging integrals

• Derivation of functions defined by integrals

• Regularization by convolution

• Ordinary differential equations

• Differential equations and systems

• Banach fixed point theorem

• Local and global Cauchy-Lipschitz theorem

• Structure of the solution space of a linear system

• Resolvent and exponential of a matrix

• Gronwall lemma

• Fourier series

• Definition: real and complex formulation

• Bessel inequality

• Riemann-Lebesgue theorem

• Dirichlet theorem

• Parseval-Plancherel theorem

• Fonction of the complex variable

• Derivability and Cauchy-Riemann equation

• Fondamental theorem of holomorphic functions

• Cauchy theorem

• Taylor and Laurent series expansion

• Residual theorem

The main concepts of the course are:

• different notions of convergence in infinite-dimensional spaces

• differential equations and systems, their structure, and what they model

• Fourier analysis

• Derivability of functions of the complex variable

Required and corequired courses

Courses requiring this course

Cours ayant celui-ci comme co-requis

Blackboard lectures

Exercise classes

Contribution to the teaching profile

This is the second part of a large course in differential and integral calculus calculus, which introduces fundamental concepts of mathematical analysis and its applications to physics.

References, bibliography, and recommended reading

Courant R. and John F. 1989 : Introduction to calculus and analysis. Springer, New York . Rudin W. : 1976. Principles of mathematical analysis. Mc Graw Hill

Contacts

Guillaume Dujardin (guillaume.dujardi@inria.fr) and Antoine Gloria (agloria@ulb.ac.be)

Method(s) of evaluation

• Other

Other

Two written exams

Two personal works

Mark calculation method (including weighting of intermediary marks)

Arithmetic average of the two exams (plus bonus from personal work).

Language(s) of evaluation

• french