MA27110 Mathematical Analysis 3
Introductory description
This is the third module in the series Analysis 1, 2, 3 that covers rigorous Analysis. It covers convergence of functions and its applications to Integration, Fourier Series and Complex Analysis.
Module aims
 Continuity, differentiability and integral of the limit of a uniformly convergent sequence of functions.
 Fourier series and their convergence.
 Foundations of Complex Analysis.
Outline syllabus
This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.
 Uniform convergence of sequences and series of functions; Weierstrass Mtest
 Application to integration: integrals of limits and series, differentiation under the integral sign
 Fourier series: convergence, Parseval, and Gibbs phenomenon (differentiability and rate of decay of coefficients)
 Complex power series and classical functions (exponential, logarithm, sine and cosine, including periodicity)
 Complex integration, contour integrals and Cauchy’s Theorem
 Applications of Cauchy’s formula to evaluate real integrals
 Laurent series, Calculus of residues
Learning outcomes
By the end of the module, students should be able to:
 Learn how to compute contour integrals: Cauchy's integral formulas and applications
 Understand uniform and pointwise convergence of functions together with properties of the limit function
 Learn the continuity, differentiability and integral of the limit of a uniformly convergent sequence of functions
 Develop working knowledge of complex differentiability (CauchyRiemann equations) and complex power series
 Develop understanding of Fourier Series including Gibbs phenomenon
Subject specific skills
 Working knowledge of series and sequences, including the development of the notions of convergence and uniform converge for sequences and series of functions.
 Good understanding of Fourier series, including their convergence, Parseval's identity and Gibbs phenomenon.
 Working knowledge of Complex Analysis, including power series, exponential and circular maps, contour integration.
 Mastery of applications of Cauchy's formula to compute integrals in R.
Transferable skills
Students will acquire key reasoning and problem solving skills which will empower them to address new problems with confidence.
Study time
Type  Required 

Lectures  20 sessions of 1 hour (20%) 
Tutorials  9 sessions of 1 hour (9%) 
Private study  71 hours (71%) 
Total  100 hours 
Private study description
71 hours private study, revision for exams, and assignments
Costs
No further costs have been identified for this module.
You do not need to pass all assessment components to pass the module.
Assessment group D
Weighting  Study time  

Assignment  15%  
Online Examination  85% 
Assessment group R
Weighting  Study time  

Inperson Examination  Resit  100%  

Feedback on assessment
Support classes, marked assignments and exam feedback.
Courses
This module is Core for:
 Year 2 of UPXAFG33 Undergraduate Mathematics and Physics (BSc MMathPhys)
 Year 2 of UPXAGF13 Undergraduate Mathematics and Physics (BSc)
 Year 2 of USTAG1G3 Undergraduate Mathematics and Statistics (BSc MMathStat)
 Year 2 of USTAGG14 Undergraduate Mathematics and Statistics (BSc)
This module is Core optional for:
 Year 2 of USTAG300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics
 Year 2 of USTAY602 Undergraduate Mathematics,Operational Research,Statistics and Economics
This module is Optional for:
 Year 3 of USTAG300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics
 Year 4 of USTAG301 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated
 Year 4 of USTAY603 Undergraduate Mathematics,Operational Research,Statistics,Economics (with Intercalated Year)