FMA_1F002_EP - Introduction to Analysis (2024-2025)

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.