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MTH2009 - Complex Analysis (2020)
MODULE TITLE | Complex Analysis | CREDIT VALUE | 15 |
---|---|---|---|
MODULE CODE | MTH2009 | MODULE CONVENER | Dr Ana Rodrigues (Coordinator) |
DURATION: TERM | 1 | 2 | 3 |
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DURATION: WEEKS | 0 | 11 | 0 |
Number of Students Taking Module (anticipated) | 200 |
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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 Analytic Number Theory (MTHM041) and 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.
Pre-requisite modules: MTH2008 (or equivalent)
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.
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.
- 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.
Scheduled Learning & Teaching Activities | 38.00 | Guided Independent Study | 112.00 | Placement / Study Abroad | 0.00 |
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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 |
Form of Assessment |
Size of Assessment (e.g. duration/length) |
ILOs Assessed |
Feedback Method |
Exercise sheets |
5 x 10 hours |
All |
Discussion at tutorials; tutor feedback on submitted answers |
Coursework | 10 | Written Exams | 90 | Practical Exams | 0 |
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Form of Assessment |
% of Credit |
Size of Assessment (e.g. duration/length) |
ILOs Assessed |
Feedback Method |
Written Exam – closed book |
90% |
2 hours |
All |
Written/verbal on request, SRS |
Coursework exercises | 10% | 2 assignments, 30 hours total | All | Annotated script and written/verbal feedback |
Original Form of Assessment |
Form of Re-assessment |
ILOs Re-assessed |
Time Scale for Re-reassessment |
All above |
Written Exam (100%) |
All |
August Ref/Def Period |
If a module is normally assessed entirely by coursework, all referred/deferred assessments will normally be by assignment.
If a module is normally assessed by examination or examination plus coursework, referred and deferred assessment will normally be by examination. For referrals, only the examination will count, a mark of 40% being awarded if the examination is passed. For deferrals, candidates will be awarded the higher of the deferred examination mark or the deferred examination mark combined with the original coursework mark
information that you are expected to consult. Further guidance will be provided by the Module Convener
Web based and Electronic Resources:
ELE: http://vle.exeter.ac.uk
Reading list for this module:
Type | Author | Title | Edition | Publisher | Year | ISBN | Search |
---|---|---|---|---|---|---|---|
Set | Stewart, I. & Tall, D. | Complex Analysis (the Hitchhiker's Guide to the Plane) | Cambridge University Press | 1983 | 000-0-521-28763-4 | [Library] | |
Set | Priestley, H.A. | Introduction to Complex Analysis | Oxford University Press | 2003 | 000-0-198-53428-0 | [Library] | |
Set | Howie, John M. | Complex Analysis | Springer | 2003 | 000-1-852-33733-8 | [Library] | |
Set | 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 | [Library] | |
Set | Rudin, R. | Principles of Mathematical Analysis | 3rd | McGraw-Hill Book Co. | 1976 | [Library] |
CREDIT VALUE | 15 | ECTS VALUE | 15 |
---|---|---|---|
PRE-REQUISITE MODULES | None |
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CO-REQUISITE MODULES | None |
NQF LEVEL (FHEQ) | 5 | AVAILABLE AS DISTANCE LEARNING | No |
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ORIGIN DATE | Wednesday 26 February 2020 | LAST REVISION DATE | Thursday 23 July 2020 |
KEY WORDS SEARCH | Complex numbers; analysis; series; functions; limits; continuity; derivatives; integration; residue; contour integral |
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