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| MODULE TITLE | Complex Analysis | CREDIT VALUE | 15 |
|---|---|---|---|
| MODULE CODE | MTH2009 | MODULE CONVENER | Dr Henri Johnston (Coordinator) |
| DURATION: TERM | 1 | 2 | 3 |
|---|---|---|---|
| DURATION: WEEKS | 0 | 11 | 0 |
| Number of Students Taking Module (anticipated) | 200 |
|---|
DESCRIPTION - summary of the module content
The central object of study in analysis is the limit and related notions of convergence, continuity, differentiation, and integration.
In this module, we carefully and rigorously develop an understanding of the analysis of functions of a complex variable. You will learn how to rigorously handle differentiation, integration, analyticity, contour integration, power series, and topology of the complex plane. Quite surprisingly, complex analysis is in many ways simpler than real analysis and has many practical applications.
The material in this module provides foundations for the study of MTHM041 Analytic Number Theory, etc. in pure mathematics as well as being the basis for many techniques for solving practical problems in economics, science, and engineering. Hence it is highly recommended to all mathematics students.
AIMS - intentions of the module
The objective of this module is to provide you with a logically based introduction to complex analysis. The primary objective is to define all the basic concepts clearly and to develop them sufficiently to provide proofs of useful theorems. This enables you to see the reason for studying analysis, and develops the subject to a stage where you can use it in a wide range of applications.
INTENDED LEARNING OUTCOMES (ILOs) (see assessment section below for how ILOs will be assessed)
On successful completion of this module, you should be able to:
Module Specific Skills and Knowledge:
1 state and prove key theorems in complex analysis using a rigorous approach;
2 understand properties of analytic functions over the complex numbers;
3 use contour integrals for computational and theoretical purposes;
Discipline Specific Skills and Knowledge:
4 apply fundamental mathematical concepts, manipulations and results in analysis;
5 formulate rigorous arguments as part of your mathematical development;
Personal and Key Transferable/ Employment Skills and Knowledge:
6 think analytically and use logical argument and deduction;
7 communicate your ideas effectively in writing and verbally;
8 manage your time and resources effectively.
SYLLABUS PLAN - summary of the structure and academic content of the module
- Epsilon-delta function limits; continuity; differentiability in the complex plane;
- Basic topology in the plane;
- Cauchy-Riemann equations; contrast to real analytic functions;
- Contour integrals; poles and singularities (isolated, removable, essential); residues; Cauchy's Theorem; Cauchy integral formulae; Taylor series and Laurent series;
- Maximum modulus principle, Liouville's theorem, fundamental theorem of algebra, meromorphic functions, residue theorem;
- Rouché’s theorem, principle of the argument;
- Applications to definite integrals, summation of series and location of zeros.
LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
| Scheduled Learning & Teaching Activities | 38.00 | Guided Independent Study | 112.00 | Placement / Study Abroad | 0.00 |
|---|
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
| Category | Hours of study time | Description |
| Scheduled Learning and Teaching Activities | 33 | Lectures including example classes |
| Scheduled Learning and Teaching Activities | 5 | Tutorials |
| Guided Independent Study | 112 | Lecture and assessment preparation; wider reading |
ASSESSMENT
FORMATIVE ASSESSMENT - for feedback and development purposes; does not count towards module grade
| Form of Assessment | Size of Assessment (e.g. duration/length) | ILOs Assessed | Feedback Method |
|---|---|---|---|
| Exercise sheets | 5 x 10 hours | 1-8 | Discussion at tutorials; tutor feedback on submitted answers |
SUMMATIVE ASSESSMENT (% of credit)
| Coursework | 20 | Written Exams | 80 | Practical Exams | 0 |
|---|
DETAILS OF SUMMATIVE ASSESSMENT
| Form of Assessment | % of Credit | Size of Assessment (e.g. duration/length) | ILOs Assessed | Feedback Method |
|---|---|---|---|---|
| Written Exam | 80 | 2 hours | 1-8 | Written/verbal on request, SRS |
| Coursework Sheets x 4 (5% each) | 20 | 30 hours (7.5 hours each) | 1-8 | Annotated script and written/verbal feedback |
DETAILS OF RE-ASSESSMENT (where required by referral or deferral)
| Original Form of Assessment | Form of Re-assessment | ILOs Re-assessed | Time Scale for Re-assessment |
|---|---|---|---|
| Written Exam | Written Exam (2 hours) (80%) | 1-8 | Referral/Deferral period |
| Coursework Sheets x 4 | Coursework exercises (4 x 5%) | 1-8 | Referral/Deferral period |
RE-ASSESSMENT NOTES
Deferrals: Reassessment will be by coursework and/or exam in the deferred element only. For deferred candidates, the module mark will be uncapped.
Referrals: Reassessment will be by a single written exam worth 100% of the module only. As it is a referral, the mark will be capped at 40%.
RESOURCES
INDICATIVE LEARNING RESOURCES - The following list is offered as an indication of the type & level of
information that you are expected to consult. Further guidance will be provided by the Module Convener
Stewart, I. & Tall, D. Complex Analysis (the Hitchhiker's Guide to the Plane). Cambridge University Press 1983 000-0-521-28763-4
Priestley, H.A. Introduction to Complex Analysis. Oxford University Press 2003 000-0-198-53428-0
Howie, John M. Complex Analysis. Springer 2003 000-1-852-33733-8
Spiegel, M.R. Schaum's outline of theory and problems of complex variables: with an introduction to conformal mapping and its appreciation. McGraw Hill 1981 000-0-070-84382-1
Rudin, R. Principles of Mathematical Analysis. 3rd Ed. McGraw-Hill Book Co. 1976
Web based and Electronic Resources:
ELE
Reading list for this module:
There are currently no reading list entries found for this module.
| CREDIT VALUE | 15 | ECTS VALUE | 7.5 |
|---|---|---|---|
| PRE-REQUISITE MODULES | None |
|---|---|
| CO-REQUISITE MODULES | None |
| NQF LEVEL (FHEQ) | 5 | AVAILABLE AS DISTANCE LEARNING | No |
|---|---|---|---|
| ORIGIN DATE | Thursday 07 March 2024 | LAST REVISION DATE | Thursday 07 March 2024 |
| KEY WORDS SEARCH | Complex numbers; analysis; series; functions; limits; continuity; derivatives; integration; residue; contour integral |
|---|
Please note that all modules are subject to change, please get in touch if you have any questions about this module.
