Module Identifier MA20110 Module Title REAL ANALYSIS Academic Year 2000/2001 Co-ordinator Dr T P McDonough Semester Semester 1 Other staff Dr K Rowlands Pre-Requisite MA11110 Mutually Exclusive MX30110 Course delivery Lecture 19 x 1 hour lectures Seminars / Tutorials 3 x 1 hour example classes Assessment Exam 2 Hours (written examination) 100% Resit assessment 2 Hours (written examination) 100%

General description
The study of real analysis is of paramount importance to any student who wishes to go beyond the routine manipulation of formulae to solve standard problems. The ability to think deductively and analyse complicated examples is essential to modify and extend concepts to new contexts. The module is geared to meet these needs.

Aims
In this module, the analytical techniques, developed in MA11110, will be extended to a more general setting. This module will provide the foundations of classical analysis in a concrete setting, with a special emphasis on applications.

Learning outcomes
On completion of this module, a student should be able to:

• determine the Fourier series of integrable functions of arbitrary period;
• apply Fourier series techniques to the summation of infinite series;
• describe the notions of continuity and differentiability for functions of several variables;
• establish whether functions of two or three variables are continuous and differentiable;
• determine whether infinite series are convergent by using varous convergence tests;
• describe the notion of uniform convergence of sequences of functions;
• use Weierstrass' M-test to test the uniform convergence of infinite series of functions;
• determine the radius of convergence of power series;
• use standard convergence theorems concerning power series;
• describe topological properties of subsets of the real line and real plane including the notion of compactness;
• state and use the Heine-Borel Theorem.

Syllabus
1. FOURIER SERIES: Convergence theorems (statements only), application of Fourier Series to sum infinite series.
2. CALCULUS OF SEVERAL VARIABLES: Continuity, differentiability, partial derivatives, higher order and mixed partial derivatives.
3. THEORY OF INFINITE SERIES: Tests for convergence, including comparison test, ratio test, integral test. Power series, radius of convergence, absolute convergence. Cauchy's principle of convergence for series.
4. UNIFORM CONVERGENCE OF SEQUENCE OF FUNCTIONS: Uniform convergence of series, Weierstrass' M-test. Cauchy's principle for uniform convergence.
5. TOPOLOGICAL CONCEPTS OF THE REAL LINE AND OF THE PLANE: Compactness, Heine-Borel Theorem.