Mathematics

MTH2001 - Analysis (2019)

MODULE TITLE CREDIT VALUE Analysis 30 MTH2001 Dr Jimmy Tseng (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11 11 0
 Number of Students Taking Module (anticipated) 244
DESCRIPTION - summary of the module content

Infinite processes appear naturally in many contexts, from science and engineering to economics. From solving the equation that finds the wave function of a quantum system in physics, processing sensor data in engineering, to calculating prices for options in economics, at the foundation of all of these are infinite processes and the pure mathematics developed to rigorously and correctly handle these processes.  That field of pure mathematics is analysis, and the central object of study in analysis is the limit which further extends to the notions of convergence, continuity, differentiation, and integration.

In this module, building on material from the first year, we will carefully and rigorously develop notions first in the context of real variables and then in the context of complex variables.  In particular, we will develop and you will learn how to rigorously handle real-variable differentiation, Riemann integration, analyticity, contour integration, power series, and the basic notions of topology. Quite surprisingly, in many ways complex analysis turns out to be more elegant than real analysis!

The material in this module will provide a foundation for the study of geometry/topology, number theory, dynamical systems, differential equations, probability, harmonic analysis, etc., in pure mathematics as well as the basis for applications in economics, science, and engineering.

Pre-requisite modules: MTH1001; MTH1002

AIMS - intentions of the module

Analysis is the theory that underpins all continuous mathematics. The objective of this module is to provide you with a logically based introduction to real and 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 analysis using a rigorous approach;

2 develop proofs related to basic topological concepts like connectedness;

3 compare and contrast the theory of analytic functions over the real and complex numbers;

4 compute contour integrals and to apply this to real analysis;

Discipline Specific Skills and Knowledge:

5 apply fundamental mathematical concepts, manipulations and results in analysis;

6 formulate rigorous arguments as part of your mathematical development;

Personal and Key Transferable/ Employment Skills and Knowledge:

7 think analytically and use logical argument and deduction;

8 communicate your ideas effectively in writing and verbally;

9 manage your time and resources effectively.

SYLLABUS PLAN - summary of the structure and academic content of the module

- Review of epsilon-N sequence limits; monotone convergence; Bolzano-Weierstrass; Cauchy sequences and completeness;

- Epsilon-delta function limits; continuity; differentiability (all in R then R^n);

- Function classes: C^k, C^infinity etc; Lipschitz continuity;

- Complex analysis; Cauchy-Riemann equations; contrast to real analytic functions;

- Inverse and implicit function theorems;

- Formal theory of Riemann integration; integrability of monotonic functions and continuous functions; problems interchanging limits in general; pointer towards Lebesgue integration and monotone/dominated convergence theorems;

- 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;

- Rouche's theorem, principle of the argument;

- Applications to definite integrals, summation of series and the location of zeros.

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study 76 224
DETAILS OF LEARNING ACTIVITIES AND TEACHING METHODS
 Category Hours of study time Description Scheduled Learning and Teaching Activities 66 Lectures including example classes Scheduled Learning and Teaching Activities 10 Tutorials Guided Independent Study 224 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 10 x 10 hours All Discussion at tutorials; tutor feedback on submitted answers
Mid-Term tests 2 x 1 hour All Feedback on marked sheets, class feedback

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams 0 100
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Written Exam A – closed book (Jan) 50 2 hours All Via SRS
Written Exam B – closed book (May) 50 2 hours All Via SRS

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 A - Closed Book Written Exam A (50%, 2hr) All August Ref/Def Period
Written Exam B - Closed Book Written Exam B (50%, 2hr) All August Ref/Def Period

RE-ASSESSMENT NOTES

In the case of module referral, the higher of the original assessment and the reassessment will be recorded for each component mark. In the case of module referral, the final mark for the module reassessment 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:

Other Resources:

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 DuChateau, P.C. Advanced Calculus Harper Collins 1992 000-0-064-67139-9 [Library]
Set McGregor, C., Nimmo, J. & Stothers, W. Fundamentals of University Mathematics 2nd Horwood, Chichester 2000 000-1-898-56310-1 [Library]
Set Gaughan, E. Introduction to Analysis 5th Thompson 1998 000-0-534-35177-8 [Library]
Set Burn, R.P. Numbers and Functions: Steps to Analysis Electronic Cambridge University Press 2005 000-0-521-41086-X [Library]
Set Bryant, V. Yet Another Introduction to Analysis Cambridge University Press 1990 978-0521388351 [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 Abbott, Stephen Understanding Analysis 2nd Springer, New York 2015 [Library]
Set Krantz, Steven G. Real Analysis and Foundations 4th CRC Press, Boca Raton, FL 2017 [Library]
Set Rudin, R. Principles of Mathematical Analysis 3rd McGraw-Hill Book Co. 1976 [Library]
CREDIT VALUE ECTS VALUE 30 15
PRE-REQUISITE MODULES MTH1001, MTH1002
NQF LEVEL (FHEQ) AVAILABLE AS DISTANCE LEARNING 5 No Tuesday 10 July 2018 Thursday 14 November 2019
KEY WORDS SEARCH Analysis; supremum; infimum; series; functions; limits; continuity; derivatives; integration; residue; contour integral