MA302-7-AU-CO:
Complex Variables
2022/23
Mathematics, Statistics and Actuarial Science (School of)
Colchester Campus
Autumn
Postgraduate: Level 7
Current
Thursday 06 October 2022
Friday 16 December 2022
15
28 April 2022
Requisites for this module
(none)
(none)
(none)
(none)
(none)
Complex numbers and functions appear in Physics (Quantum Mechanics, especially to solve Schrödinger equation to find the wave function), in Engineering (Control theory and Signal Analysis as the Fourier transform requires integrating complex valued functions).
This module is a continuation of the module MA114 where Complex Numbers briefly is introduced. This module covers in details complex functions and variables and expands on complex derivatives and integration, and more advanced topics.
To introduce functions of a complex variable and techniques for complex integration including Cauchy’s theorem, integral formula, residue formula, and Jordan’s Lemma, and mappings of elementary functions and conformal mapping (including Riemann Mapping Theorem and Möbius transformation).
On successful completion of the course, students should be able to:
1. Express complex numbers in both Cartesian and polar forms;
2. Parametrize curves and plot regions in the complex plane
3. Carry out calculations of limits, continuity, and differentiability of complex functions.
4. Determine whether and where a function is holomorphic / analytic;
5. Carry out complex integration via line integrals, Cauchy’s Theorem, Cauchy’s integral formula and Cauchy’s differentiation formula
6. Obtain appropriate series expansions of functions;
7. Evaluate residues at pole singularities;
8. Apply the Residue Theorem to the calculation of real integrals.
9. Find images regions and curves under mapping of elementary and certain functions.
10. Find transformations which maps certain regions to certain regions.
Complex numbers:
Cartesian and polar forms
Lines, circles and regions in the complex plane
Functions of a complex variable:
Limits
Continuity
Derivatives
Holomorphic/analytic functions
Cauchy-Riemann Equations
Elementary functions
The exponential function
The logarithmic function
Branch points and Branch cuts
Complex Exponents
Trigonometric and hyperbolic functions
Elementary functions as mapping
Complex Integration:
Line integrals
Cauchy's theorem
Cauchy's integral formula
Derivatives of an analytic function (Cauchy's differentiation formula)
Consequences of Cauchy's Integral formula (Liouville's theorem, Maximum Modulus Principle, Fundamental theorem of algebra)
Morera's Theorem
Sequences and Series of Complex Numbers:
Taylor series
Analytic functions and their relationship to holomorphic functions
Laurent's theorem
Residue and Poles
Calculation of residues
Cauchy's residue theorem
Argument principle
Rouch\'{e}'s theorem.
Jordan's lemma
Calculation of definite integrals using residue theory.
Level 7 Only
Conformal Mappings and Applications
Linear Mappings,
Mappings by 1/z
Automorphisms of the Riemann sphere: Möbius transformation
Conformal Mappings
Transformation from certain regions to certain regions
Riemann Mapping Theorem
Applications of Conformal Mapping
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: In-Person, Open Book (Restricted), 120 minutes during Summer (Main Period)
|
| Exam |
Reassessment Main exam: In-Person, Open Book (Restricted), 120 minutes during September (Reassessment 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
Yes
No
Yes
Prof Stephen Langdon
Brunel University London
Professor
Available via Moodle
Of 38 hours, 38 (100%) hours available to students:
0 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.