Study information

# Complex Analysis - 2024 entry

MODULE TITLE CREDIT VALUE Complex Analysis 15 MTH2009 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 Guided Independent Study Placement / Study Abroad 38 112 0
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 Written Exams Practical Exams 20 80 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:

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