## MA2RA2-Real Analysis II

Module Provider: Mathematics and Statistics
Number of credits: 10 [5 ECTS credits]
Level:5
Terms in which taught: Autumn / Summer term module
Pre-requisites: MA1RA1 Real Analysis I or MA2RA1 Real Analysis I and MA1FM Foundations of Mathematics and MA1CA Calculus and MA1LA Linear Algebra
Non-modular pre-requisites:
Co-requisites:
Modules excluded: MA2RCA Real and Complex Analysis or MA3RCA Real and Complex Analysis
Current from: 2021/2

Module Convenor: Dr Nikos Katzourakis

Type of module:

Summary module description:
This module continues the study of analysis to the point where it relates to topics studied in other courses, such as integration and differentiation.

Aims:
To introduce students to rigorous analysis, enable them to relate this to calculus and demonstrate that analysis can produce some worthwhile results which calculus cannot.

Assessable learning outcomes:
By the end of the module students are expected to be able to:

- use the ideas of integration and differentiation in a rigorous way

- manipulate infinite series and apply them in problems involving differentiation and integration

- understand that there are situations in which the order of limiting operations may not be inverted, and justify some simple cases where the inversion is legitimate.

Students will encounter situations where existence and uniqueness of solutions may be proved where the solutions cannot be expressed explicitly.

Outline content:
This module continues the study of analysis to the point where it relates to topics studied in other courses, such as integration and differentiation. Some of this material is just the logical underpinning of the results one would expect, but there are some surprises. This is particularly evident in the section on uniform convergence, where attention is given to interchanging the order of limiting processes, a useful technique but one which is not always valid. The course will conclude with some applications of analysis to other areas producing results which could not be obtained without the use of analysis.

Brief description of teaching and learning methods:
Lectures, supported by problem sheets and lecture-based tutorials.

Contact hours:
 Autumn Spring Summer Lectures 20 2 Tutorials 10 Guided independent study: 68 Total hours by term 98 2 Total hours for module

Summative Assessment Methods:
 Method Percentage Written exam 80 Set exercise 20

Summative assessment- Examinations:
2 hours

Summative assessment- Coursework and in-class tests:
Assignments and one examination paper.

Formative assessment methods:
Problem sheets

Penalties for late submission:

The Support Centres will apply the following penalties for work submitted late:

• where the piece of work is submitted after the original deadline (or any formally agreed extension to the deadline): 10% of the total marks available for that piece of work will be deducted from the mark for each working day (or part thereof) following the deadline up to a total of five working days;
• where the piece of work is submitted more than five working days after the original deadline (or any formally agreed extension to the deadline): a mark of zero will be recorded.
The University policy statement on penalties for late submission can be found at: http://www.reading.ac.uk/web/FILES/qualitysupport/penaltiesforlatesubmission.pdf
You are strongly advised to ensure that coursework is submitted by the relevant deadline. You should note that it is advisable to submit work in an unfinished state rather than to fail to submit any work.

Assessment requirements for a pass:
A mark of 40% overall.

Reassessment arrangements:
One examination paper of 2 hours duration in August/September - the resit module mark will be the higher of the exam mark (100% exam) and the exam mark plus previous coursework marks (80% exam, 20% coursework).