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Module Specifications

Archived Version 2020 - 2021

Module Title
Module Code

Online Module Resources

NFQ level 8 Credit Rating 7.5
Pre-requisite None
Co-requisite None
Compatibles None
Incompatibles None

This module will cover sequences and series of functions and introduce several different notions of convergence and their properties. These topics are seen to have a natural formulation in linear algebra with vector spaces of functions. Key examples will be power series, Fourier series and Fourier transforms. Some applications of these 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 Fourier transforms to solve ordinary linear differential equations with constant coefficients

Workload Full-time hours per semester
Type Hours Description
Lecture36Students 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.
Tutorial12Students will show their solutions to homework questions and will receive help with and feed-back on these solutions.
Independent Study78Corresponding 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: 126

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

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.

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.

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.

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% Examination Weight%
Course Work Breakdown
TypeDescription% of totalAssessment Date
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
Indicative Reading List

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    Programme or List of Programmes