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

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