MA2CANU-Complex Analysis

Module Provider: Mathematics and Statistics
Number of credits: 10 [5 ECTS credits]
Level:5
Terms in which taught: Spring term module
Pre-requisites: MA0FMNU Foundations of Mathematics and MA0MANU Mathematical Analysis and MA1RA1NU Real Analysis 1 and MA1RA2NU Real Analysis II and MA1LANU Linear Algebra
Non-modular pre-requisites:
Co-requisites:
Modules excluded:
Current from: 2021/2

Module Convenor: Dr Jochen Broecker
Email: j.broecker@reading.ac.uk

Type of module:

Summary module description:

This module provides an introduction to complex analysis. The students will learn something new and see how the applications of complex analysis reinforce the key ideas. The module moves gradually and steadily from complex numbers to complex-valued functions to integrals and series. The principles students learn in the class could be helpful for them in subsequent courses.Ìý

The Module lead at NUIST is Dr Jian Ding.

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Aims:

To introduce students to complex analysis 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:

• solve problems involving holomorphic functions;


• recognise 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:

Differentiable functions of a complex variable are remarkably well-behaved, and most of the technical complications of the real case do not 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 theor em of Cauchy established, from which a whole range of applications follow. Some applications to the evaluation of real integrals are given.

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Brief description of teaching and learning methods:

Lectures supported by problem sheets and lecture-based tutorials.

Contact hours:

Ìý	Autumn	Spring	Summer
Lectures		40
Tutorials		8
Guided independent study:	Ìý	Ìý	Ìý
Ìý Ìý Wider reading (independent)		30
Ìý Ìý Wider reading (directed)		12
Ìý Ìý Exam revision/preparation		10
Ìý	Ìý	Ìý	Ìý
Total hours by term	0		0
Ìý	Ìý	Ìý	Ìý
Total hours for module	100

Summative Assessment Methods:

Method	Percentage
Written exam	80
Class test administered by School	20

Summative assessment- Examinations:

2 hours.

Summative assessment- Coursework and in-class tests:

One examination and a number of class tests.

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:
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):

Last updated: 2 November 2021