MA302-6-AU-CO:
Complex Variables and Applications
2021/22
Mathematics, Statistics and Actuarial Science (School of)
Colchester Campus
Autumn
Undergraduate: Level 6
Current
Thursday 07 October 2021
Friday 17 December 2021
15
19 August 2021
Requisites for this module
(none)
(none)
(none)
(none)
(none)
BSC G100 Mathematics,
BSC G102 Mathematics (Including Year Abroad),
BSC G103 Mathematics (Including Placement Year),
BSC G104 Mathematics (Including Foundation Year),
MMATG198 Mathematics,
BSC G1G4 Mathematics with Computing (Including Year Abroad),
BSC G1G8 Mathematics with Computing (Including Foundation Year),
BSC G1GK Mathematics with Computing,
BSC G1IK Mathematics with Computing (Including Placement Year),
BSC G1F3 Mathematics with Physics,
BSC G1F4 Mathematics with Physics (Including Placement Year),
BSC G1F5 Mathematics with Physics (Including Foundation Year),
BSC GCF3 Mathematics with Physics (Including Year Abroad),
MSCIG199 Mathematics and Data Science
An introduction to complex analysis, up to and including evaluation of contour integrals using the Residue theorem.
To introduce functions of a complex variable and techniques for complex integration including Cauchy’s theorem, integral formula, residue formula, and Jordan’s Lemma.
On successful completion of the course, students should be able to:
1. Express complex numbers in both Cartesian and polar forms;
2. Identify curves and regions in the complex plane defined by simple formulae;
3. Determine whether and where a function is holomorphic / analytic;
4. Carry our complex integration via line integrals, Cauchy’s Theorem, Cauchy’s integral formula and Cauchy’s differentiation formula.
5. Obtain appropriate series expansions of functions;
6. Evaluate residues at pole singularities;
7. Apply the Residue Theorem to the calculation of real integrals.
Syllabus
Complex numbers:
- Cartesian and polar forms
- Lines, circles and regions in the complex plane
Functions of a complex variable:
- derivatives
- holomorphic functions
- Cauchy-Riemann Equations
Complex Integration:
- Line integrals
- Cauchy's theorem
- Cauchy's integral formula
- Derivatives of an analytic function (Cauchy's differentiation formula)
Liouville's theorem
Sequences and Series of Complex Numbers:
- Taylor series
- Analytic functions and their relationship to holomorphic functions
- Laurent's theorem
Residue Integration Methods:
- Calculation of residues at poles
- Cauchy's residue theorem
Argument principle, Rouch\'{e}'s theorem, examples. Fundamental Theorem of Algebra.
- Jordan's lemma
- Calculation of definite integrals using residue theory.
Teaching will be delivered in a way that blends face-to-face classes, for those students that can be present on campus, with a range of online lectures, teaching, learning and collaborative support.
This module does not appear to have a published bibliography for this year.
Assessment items, weightings and deadlines
| Coursework / exam |
Description |
Deadline |
Coursework weighting |
| Coursework |
Assignment 1 |
|
|
| Coursework |
Assignment 2 |
|
|
| Exam |
Main exam: 180 minutes during Summer (Main Period)
|
Exam format definitions
- Remote, open book: Your exam will take place remotely via an online learning platform. You may refer to any physical or electronic materials during the exam.
- In-person, open book: Your exam will take place on campus under invigilation. You may refer to any physical materials such as paper study notes or a textbook during the exam. Electronic devices may not be used in the exam.
- In-person, open book (restricted): The exam will take place on campus under invigilation. You may refer only to specific physical materials such as a named textbook during the exam. Permitted materials will be specified by your department. Electronic devices may not be used in the exam.
- In-person, closed book: The exam will take place on campus under invigilation. You may not refer to any physical materials or electronic devices during the exam. There may be times when a paper dictionary,
for example, may be permitted in an otherwise closed book exam. Any exceptions will be specified by your department.
Your department will provide further guidance before your exams.
Overall assessment
Reassessment
Module supervisor and teaching staff
Dr Murat Akman, email: murat.akman@essex.ac.uk.
Dr Murat Akman
murat.akman@essex.ac.uk
No
No
No
Prof Stephen Langdon
Brunel University London
Professor
Available via Moodle
Of 60 hours, 57 (95%) hours available to students:
3 hours not recorded due to service coverage or fault;
0 hours not recorded due to opt-out by lecturer(s), module, or event type.
Disclaimer: The University makes every effort to ensure that this information on its Module Directory is accurate and up-to-date. Exceptionally it can
be necessary to make changes, for example to programmes, modules, facilities or fees. Examples of such reasons might include a change of law or regulatory requirements,
industrial action, lack of demand, departure of key personnel, change in government policy, or withdrawal/reduction of funding. Changes to modules may for example consist
of variations to the content and method of delivery or assessment of modules and other services, to discontinue modules and other services and to merge or combine modules.
The University will endeavour to keep such changes to a minimum, and will also keep students informed appropriately by updating our programme specifications and module directory.
The full Procedures, Rules and Regulations of the University governing how it operates are set out in the Charter, Statutes and Ordinances and in the University Regulations, Policy and Procedures.