## MA4CA2-Complex Analysis II

Module Provider: Mathematics and Statistics
Number of credits: 10 [5 ECTS credits]
Level:7
Terms in which taught: Spring term module
Pre-requisites: MA2RCA Real and Complex Analysis
Non-modular pre-requisites:
Co-requisites:
Modules excluded: MA3CA2 Complex Analysis II
Current from: 2018/9

Module Convenor: Dr Horatio Boedihardjo

Type of module:

Summary module description:

This module continues the study of functions of one complex variable.

Aims:

To continue the study of functions of one complex variable, and in particular of analyticity and its consequences.

Assessable learning outcomes:

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

• Deduce further properties of analytic functions from classical theorems;

• Identify different type of singularities of complex functions;

• Apply standard results about singularities of complex functions to prove further analytic results;

• Use residue calculus to study zeros and poles of given functions.;

• Use residues to compute complex and real integrals;

• Use Rouche's Theorem to estimate the number of zeros of complex functions in regions of the complex plane;

• State, prove and apply the Identity theorem and Maximum Modulus Theorem.

This module will be assessed to a greater depth than the excluded module MA3CA2.

• The use of complex analytic arguments to derive basic results in algebra and real analysis;

• Provide rigorous proofs for results in complex analysis.

Outline content:

Starting from Cauchy’s theorem and its consequences, discussed in the prerequisite complex analysis module (MA2RCA), the module will present further properties of sequences and series of complex functions, and singularities of complex functions, leading to the concept of analytic continuation.

More advanced consequences of residue calculus, such as the maximum modulus, argument principle and Rouche’s theorem, will be presented.

Brief description of teaching and learning methods:

Lectures supported by problem sheets.

Contact hours:
 Autumn Spring Summer Lectures 20 Tutorials 10 Guided independent study 70 Total hours by term 100.00 Total hours for module 100.00

Summative Assessment Methods:
 Method Percentage Written exam 100

Summative assessment- Examinations:

2 hours.

Summative assessment- Coursework and in-class tests:

Formative assessment methods:

Problem sheets.

Penalties for late submission:
The Module Convener 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 50% overall.

Reassessment arrangements:

One examination paper of two hours duration in August/September.