Bachelor 1

Analysis (MAA 102) is an introductory-level mathematical analysis course which provides a well-balanced approach between foundational notions and calculus. It is designed to equip students with the fundamental analytical tools required to pursue studies in Mathematics and, more generally, in any scientific field (Physics, Mechanics, Economics, Engineering, etc).

The objective is to present fundamental notions and results regarding the set of real and complex numbers, real and complex-valued sequences, real and complex-valued infinite series and functions of one real variable.

With respect to the expected initial knowledge of the students, the Course follows a more systematic approach, providing a few insights on the roots of analysis and proving all important results. Though in the continuity of the students' previous studies in Mathematics, this course may also be a turning point towards more rigor and proofs.

In particular, this course covers many aspects of the theory of real valued and complex valued sequences, including the notion of subsequence, of accumulation points of a sequence and the Bolzano-Weierstrass Theorem. Building on this, the course also covers the theory of real valued and complex valued infinite series, including the study of absolutely convergent series and alternating sequences.

Next, the course also adresses the study of real valued continuous functions of one variable, starting from the definition up to global properties of continuous functions such as the Intermediate Value Theorem.

The third part of the course is concerned with the differentiability of real valued functions of one variable including higher differentiability. This leads to the Mean Value Theorem and Inverse Function Theorem. The study of classical functions (trigonometric functions, hyperbolic functions, etc) is also presented.

The course ends with an introduction to the theory of approximation of differentiable functions by polynoms and in particular the Taylor-Lagrange Theorem which provides the Taylor series of many interesting functions.

Integrals and differential calculus (MAA105) develop students's skills in crucial analytical tools, in particular integration theory. The approach to integration employed in this course is Riemann's integral, a foundational mathematical theory. This course also introduces students to two important and related topics covered in the Bachelor program: Taylor expansions (a tool for function approximation) and differential equations, which are required to understand basic physical problems (trajectories, populations, etc.)

The first part of this course focusses on the notion of Riemann integral. After introducing the notion of Riemann integrable function, we briefly discuss the basic properties of such functions. Next we present the classical methods for computing integrals (integration by parts, integration by substitution, integration of rational fractions, elementary abelian integrals…).

The second part is dedicated to Taylor expansions. We review Taylor formulas for approximation of functions near a given point, then present the theory of Taylor expansion, giving all the tools for computing them in practice, as well as their direct applications.

The third part is the study of ordinary differential equations, mainly first order linear differential equations and linear systems of ODEs, with a special focus on linear differential equations with constant coefficients.