Registry
Module Specifications
Archived Version 2016  2017
 
Description This module will introduce students to the notions of sequences and series of functions and various forms of their convergence will be discussed. 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  
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. Pointwise convergence, uniform convergence and properties assured by uniform convergence. The Weierstrass' Mtest. The Weierstrass Approximation Theorem. Uniform continuity, Bernstein polynomials. CONVERGENCE IN NORM The L^2Norm, 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 pointwise 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.  
 
Indicative Reading List  
Other Resources None  
Programme or List of Programmes  
Archives: 
