Registry

Module Specifications

Current Academic Year 2012 - 2013
Please note that this information is subject to change.

Module Title	Analysis II
Module Code	MS229
School	School of Mathematics
Online Module Resources
Module Co-ordinator	Semester 1: Michael Clancy (maths) Semester 2: Michael Clancy (maths) Autumn: Michael Clancy (maths)
Module Teacher	Michael Clancy (maths)
NFQ level	8	Credit Rating	5
Pre-requisite	None
Co-requisite	None
Compatibles	None
Incompatibles	None

Description

This module will introduce students to the notions of sequences & series of functions and various forms of their convergence will be discuss. General Fourier series will be presented as a natural extension of linear algebra to infinite dimensions highlighting the geometric aspects of the theory. The nature of the convergence (with selected proofs) of general and trigonometric Fourier series will be explored. Applications of the various topics will also be considered. Students will attend lectures on the course material and will work, independently, to solve problems on topics related to the course material. The students will have an opportunity to review their solutions, with guidance, at weekly tutorials.

Learning Outcomes

1. Determine the nature of convergence of selected sequences and series of functions
2. State selected definitions and theorems
3. Calculate the trigonometric Fourier series of elementary functions and be able to use such series to sum series of real numbers
4. Apply the theory of Laplace transforms to solve systems of ordinary linear differential equations
5. Prove selected theorems

Workload	Full-time hours per semester
*Type*	*Hours*	*Description*
Lecture	36	Students will attend lectures where new material will be presented and explained. Also attention will be drawn to various supporting material and tutorials as the course progresses.
Tutorial	12	Students will show their solutions to homework questions and will receive help with and feed-back on these solutions.
Independent learning	77	Corresponding to each lecture students will devote approximately one additional hour of independent study to the material discussed in that lecture or to work on support material when attention is drawn to such in lectures. Before each tutorial students will devote approximately three and a half hours to solving homework problems which are to be discussed in that tutorial.
Total Workload: 125

All module information is indicative and subject to change. For further information,students are advised to refer to the University's Marks and Standards and Programme Specific Regulations at: http://www.dcu.ie/registry/examinations/index.shtml

Indicative Content and Learning Activities

SEQUENCES AND SERIES.
Review of sequences of Real/Complex numbers and the Cauchy condition for convergence. Point-wise convergence, uniform convergence and properties assured by uniform convergence. The Weierstrass' M-test. The Weierstrass Approximation Theorem. Uniform continuity, Bernstein polynomials..

CONVERGENCE IN NORM.
The L^2-Norm, complete sets and orthonormal sets of functions. Fourier coefficients and general Fourier series. Optimal approximation property of truncated Fourier series. Bessel's inequality and Parseval's identity..

TRIGONOMETRIC FOURIER SERIES.
Piecewise continuous functions and the completeness of trigonometric system. Conditions for point-wise convergence of trigonometric Fourier series. Calculation of Fourier series. Sine series and cosine series. Differentiation of a trigonometric Fourier series. Complex form of trigonometric Fourier Series..

FOURIER AND LAPLACE TRANSFORMS.
The Fourier transform as a limiting form of the Fourier series. The Fourier inversion formula. The Laplace transform of exponentially dominated functions. Properties of the Laplace transform and its application to systems of linear ODE's..

Assessment Breakdown
Continuous Assessment	25%	Examination Weight	75%

Course Work Breakdown
Type	Description	% of total	Assessment Date
Oral presentations	In-class test	25%	n/a

Reassessment Requirement
Resit arrangements are explained by the following categories; 1 = A resit is available for all components of the module 2 = No resit is available for 100% continuous assessment module 3 = No resit is available for the continuous assessment component
This module is category 3

Indicative Reading List

Churchill, R. V. and Brown, J. W.: 0, Fourier Series and Boundary Value Problems, McGrawHill,
KÅ‘rner, T. W: 0, Fourier Analysis, Cambridge University press,

Other Resources

None

Array

Programme or List of Programmes

ACM

BSc Actuarial Mathematics

CAFM

Common Entry into Mathematical Sciences

BSc in Financial & Actuarial Mathematics

SHSA

Study Abroad (Science & Health)

SHSAO