Mathematics

MTH2009 - Complex Analysis (2022)

Back | Download as PDF
MODULE TITLEComplex Analysis CREDIT VALUE15
MODULE CODEMTH2009 MODULE CONVENERDr Jimmy Tseng (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 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)

 

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

All

Discussion at tutorials; tutor feedback on submitted answers

 

SUMMATIVE ASSESSMENT (% of credit)
Coursework 10 Written Exams 90 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 – closed book

90%

2 hours (Summer)

All

Written/verbal on request, SRS

Coursework exercises 10% 30 hours total All 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-reassessment

Written Exam*

Written Exam (2 hours) (90%)

All

August Ref/Def Period

Coursework Exercises* Coursework exercises (10%)  All August Ref/Def Period

*Please refer to reassessment notes for details on deferral vs. Referral reassessment

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

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 MTH2008
CO-REQUISITE MODULES
NQF LEVEL (FHEQ) 5 AVAILABLE AS DISTANCE LEARNING No
ORIGIN DATE Wednesday 26 February 2020 LAST REVISION DATE Thursday 04 August 2022
KEY WORDS SEARCH Complex numbers; analysis; series; functions; limits; continuity; derivatives; integration; residue; contour integral