MA0221: Analysis III

School Cardiff School of Mathematics
Department Code MATHS
Module Code MA0221
External Subject Code G100
Number of Credits 10
Level L5
Language of Delivery English
Module Leader Professor Karl Schmidt
Semester Autumn Semester
Academic Year 2014/5

Outline Description of Module

This module follows up on the introduction to mathematical analysis provided in the first-year. It develops further important basic concepts of analysis, including convergence of functional sequences and series, the interchangeability (or otherwise) of limits, uniform continuity, as well as their applications to the study of functions defined through series, integrals or differential equations. These concepts form the foundation for later courses on Complex Analysis, Differential Equations and Fourier and Functional Analysis.

Prerequisite Modules: MA0123 Analysis I, MA0126 Analysis II

On completion of the module a student should be able to

  • Know firmly all basic definitions, including those of real numbers, supremum and infimum, limit of a sequence, sum of a series, pointwise and uniform convergence of functional sequences and series,continuity, uniform continuity, differentiability, integrability of functions
  • Explain the logical foundations and mathematical ideas in the development of mathematical analysis, including knowledge of the precise formulations of the main theorems and of their proofs
  • Apply the theoretical results  to studying the properties of concrete functions, sequences and series

How the module will be delivered

27 - 50 minute lectures

Some handouts will be provided in hard copy or via Learning Central, but students will be expected to take notes of lectures.

Students are also expected to undertake at least 50 hours private study including preparation of solutions to given exercises.

Skills that will be practised and developed

  • Correct interpretation and manipulation of quantified logical expressions
  • Ability to understand and assimilate medium-length mathematical arguments
  • Ability to apply techniques of proofs from a larger toolbox, appreciating the fact that generally there is no single approach covering all problems
  • Ability to recognise the convergence properties of functional sequences and series and the analytical properties of functions defined via limiting processes
  • Appreciation of the necessity to justify the interchange of limit processes, and ability to provide such justification
  • Ability to apply theoretical results to concrete cases
  • Use of mathematical language in a logically and rhetorically correct way

How the module will be assessed

Formative assessment is carried out by means of regular homeworks.  Feedback to students on their progress towards learning outcomes is provided by checking homeworks and during exercise lectures.  

The summative assessment is the written examination at the end of the module.  This gives students the opportunity to demonstrate their overall achievement of learning outcomes.  It also allows them to give evidence of the higher levels of knowledge and understanding required for above average marks.

The examination paper has two sections of equal weight.  Section A contains a number of compulsory questions of variable length but normally short, including main definitions, simple applications and some basic theorems.  Section B has a choice of two from three equally weighted questions.  It contains further, more sophisticated applications and also formulations and/or proofs of some of the theorems in covered in lectures.

Assessment Breakdown

Type % Title Duration(hrs)
Exam - Autumn Semester 100 Analysis Iii 2

Syllabus content

  • Elements of propositional calculus, completeness of the set of real numbers, the Bolzano-Weierstrass theorem, Cauchy sequences;
  • pointwise and uniform convergence of functional sequences and series, analytical properties of limit functions, applications;
  • fundamental property and radius of convergence of power series, Abel's limit theorem;
  • uniform continuity of functions, integrability, continuity and differentiability of integrals with respect to a parameter

Essential Reading and Resource List

Please see Background Reading List for an indicative list.

Background Reading and Resource List

Real Analysis, Howie, J. M., Springer, 2000

Elementary Mathematical Analysis, Clark., C., Wadsworth, 1982


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