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Throughout the 2021-22 academic year, we will be prioritising face to face teaching as part of a blended learning approach that builds on the lessons learned over the course of the Coronavirus pandemic. Teaching will vary between online and on-campus delivery through the year, and you should read guidance from the academic department for details of how this will work for a particular module. You can find out more about the University’s overall response to Coronavirus at: https://warwick.ac.uk/coronavirus.

MA271-10 Mathematical Analysis 3

Department
Warwick Mathematics Institute
Level
Undergraduate Level 2
Module leader
Roger Tribe
Credit value
10
Module duration
10 weeks
Assessment
Multiple
Study location
University of Warwick main campus, Coventry
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 web page

Module aims
  1. Continuity, differentiability and integral of the limit of a uniformly convergent sequence of functions.
  2. Fourier series and their convergence.
  3. 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 M-test
  • 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 (Cauchy-Riemann 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
In-person Examination - Resit 100%
  • Answerbook Gold (24 page)
Feedback on assessment

Support classes, marked assignments and exam feedback.

Past exam papers for MA271

Courses

This module is Core for:

  • Year 2 of UPXA-FG33 Undergraduate Mathematics and Physics (BSc MMathPhys)
  • Year 2 of UPXA-GF13 Undergraduate Mathematics and Physics (BSc)
  • Year 2 of USTA-G1G3 Undergraduate Mathematics and Statistics (BSc MMathStat)
  • Year 2 of USTA-GG14 Undergraduate Mathematics and Statistics (BSc)

This module is Core optional for:

  • Year 2 of USTA-G300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics
  • Year 2 of USTA-Y602 Undergraduate Mathematics,Operational Research,Statistics and Economics

This module is Optional for:

  • Year 3 of USTA-G300 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics
  • Year 4 of USTA-G301 Undergraduate Master of Mathematics,Operational Research,Statistics and Economics (with Intercalated
  • Year 4 of USTA-Y603 Undergraduate Mathematics,Operational Research,Statistics,Economics (with Intercalated Year)