Module Identifier MA20110
Module Title REAL ANALYSIS
Co-ordinator Dr Robert J Douglas
Semester Semester 1
Pre-Requisite MA11110
Mutually Exclusive MX30110
Course delivery Lecture   19 x 1 hour lectures
Seminars / Tutorials   3 x 1 hour example classes
Assessment
Assessment TypeAssessment Length/DetailsProportion
Semester Exam2 Hours (written examination)  100%
Supplementary Assessment2 Hours (written examination)  100%

#### 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 more variables are continuous and differentiable;
• determine whether infinite series are convergent by using various convergence tests;
• describe the notion of uniform convergence of sequences of functions;
• use the 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.

#### Brief 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.

#### Content

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. The Cauchy principle of convergence for series.
4. UNIFORM CONVERGENCE OF SEQUENCE OF FUNCTIONS: Uniform convergence of series, the Weierstrass M-test. The Cauchy principle for uniform convergence.