# Mathematics

## MTH2009 - Complex Analysis (2022)

MODULE TITLE CREDIT VALUE Complex Analysis 15 MTH2009 Dr 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 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 All Discussion at tutorials; tutor feedback on submitted answers

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams Practical Exams 10 90 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