MA2RCA-Real and Complex Analysis

Module Provider: Mathematics and Statistics
Number of credits: 20 [10 ECTS credits]
Level:5
Terms in which taught: Autumn / Spring / Summer module
Pre-requisites: MA1FM Foundations of Mathematics MA1CA Calculus MA1LA Linear Algebra MA1RA1 Real Analysis I
Non-modular pre-requisites:
Co-requisites:
Modules excluded: MA3RCA Real and Complex Analysis
Current from: 2020/1

Module Convenor: Dr Titus Hilberdink

Email: t.w.hilberdink@reading.ac.uk

Type of module:

Summary module description:
The first part of this module continues the study of analysis to the point where it relates to topics in other courses, such as integration and differentiation. The second part provides an introduction to complex analysis.

Aims:
To further deepen students’ understanding of real analysis and introduce students to complex analysis. To enable students to relate this to calculus and demonstrate that analysis can produce remarkable results that calculus cannot, and enable them to use complex variable techniques, particularly in some cases where the original problem does not involve complex numbers.

Assessable learning outcomes:

By the end of the module, students are expected to be able  to:

• use the concepts of continuity, differentiation and integration in a rigorous  way;


• determine whether a sequence or series of functions converges uniformly;


• manipulate infinite series and apply them in problems involving continuity, differentiation and integration;


• solve problems involving holomorphic  functions;


• recogn ize and be able to apply the complex exponential and  logarithm;


• evaluate path integrals of complex  functions;


• identify singularities and residues of holomorphic  functions;


• calculate appropriate real integrals using complex  techniques.

Additional outcomes:
By the end of the module the student will begin to understand and recognise some of the structure of holomorphic functions.

Outline content:
The definition of the Riemann integral and Riemann sums. The properties of the integral. Integrability of continuous functions. Fundamental theorem of calculus. Sequences and series of functions. Uniform convergence. Power series. Special functions, such as exponential, logarithmic and trigonometric functions. Applications of real analysis.


Differentiable functions of a complex variable are remarkably well-behaved, and most of the technical complications of the real case do n ot arise with complex functions. This leads to some remarkably powerful results, and it turns out that complex variable techniques often offer the simplest method of evaluating certain real integrals. The notion of complex differentiability relates closely with power series. Contour integration in the complex plane will be introduced and the remarkable theorem of Cauchy established, from which a whole range of applications follow. Some applications to the evaluation of real integrals are given.< br />

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

Contact hours:

	Autumn	Spring	Summer
Lectures	20	20	4
Tutorials	10	10
Guided independent study:	68	68
Total hours by term	98	98	4
Total hours for module	200

Summative Assessment Methods:

Method	Percentage
Written exam	80
Set exercise	20

Summative assessment- Examinations:

3 hours.

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

Formative assessment methods:
Problem sheets.

Penalties for late submission:

The Module Convenor 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[1] (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 three 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).

Additional Costs (specified where applicable):
1) Required text books:
2) Specialist equipment or materials:
3) Specialist clothing, footwear or headgear:
4) Printing and binding:
5) Computers and devices with a particular specification:
6) Travel, accommodation and subsistence:

Last updated: 4 April 2020