|Module Title||REAL ANALYSIS|
|Co-ordinator||Dr T P McDonough|
|Course delivery||Lecture||19 x 1hour lectures|
|Seminars / Tutorials||3 x 1hour example classes|
|Assessment||Exam||2 Hours (written examination)||100%|
|Resit assessment||2 Hours (written examination)||100%|
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.
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.
On completion of this module, a student should be able to:
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.
** Recommended Text
R Haggarty. Fundamentals of Mathematical Analysis. Addison-Wesley
J Marsden & M Hoffman. Elementary Classical Analysis. Freeman
** Supplementary Text
R Bartle & D Sherbert. Introduction to Real Analysis. Wiley
** Reference Text
W Parzynski & P Zipse. Introduction to Mathematical Analysis. McGraw-Hill