# Mathematics

## ECM2701 - Analysis (2015)

MODULE TITLE CREDIT VALUE Analysis 15 ECM2701 Prof Mohamed Saidi (Coordinator)
DURATION: TERM 1 2 3
DURATION: WEEKS 11 weeks 0 0
 Number of Students Taking Module (anticipated) 196
DESCRIPTION - summary of the module content

This module will provide you with knowledge of fundamental mathematical concepts, manipulations and results in analysis, and with an appreciation of the need for rigour in their development. Both of these are required for progression and success in future study of calculus based subject material and its applications.  You will study properties of the real number system; limits of functions and continuity; continuous functions; differentiation and integration of complex functions. By the end of this course, you will have developed your analytical thought processes and gained experience in logical argument and deduction.

Prerequisite module: ECM1702 or equivalent

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 analysis and some aspects of 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 demonstrate understanding of the fundamentals of analysis using a rigorous approach.
Discipline Specific Skills and Knowledge:
2 show knowledge of fundamental mathematical concepts, manipulations and results in analysis;
3 appreciate the need for rigour in your development.
Personal and Key Transferable/ Employment Skills and  Knowledge:
4 think analytically and use logical argument and deduction.

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

- sequences: convergence of (real or complex) sequences; elementary properties of sequences;

- properties of the real number system: completeness;

- proofs of elementary propositions on sup and inf; the convergence of bounded monotone sequences; the Bolzano-Weierstrass Theorem that every bounded real
sequence has a convergent subsequence;

- limits of functions and continuity: the definition of a limit of a (real or complex) function at a point; elementary properties of limits; examples; convergence and absolute convergence of (real or complex) infinite series, including power series and Fourier series;

- continuous functions: definition; chain rule; theorem that a continuous function defined on a closed bounded interval is bounded and attains its bounds; intermediate value theorum;

- differentiation: definition of derivative for real or complex functions; elementary rules for differentiation and chain rule; differentiation of a convergent power series; Rolle's theorem; first and second mean value theorems and applications to problems involving specific functions;

- integration of a complex function: definition of path integral;

- basic properties of complex integrations.

LEARNING AND TEACHING
LEARNING ACTIVITIES AND TEACHING METHODS (given in hours of study time)
 Scheduled Learning & Teaching Activities Guided Independent Study Placement / Study Abroad 44 106 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 11 Tutorials Guided independent study 106 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
Fortnightly exercise 40 hours overall All Discussion at tutorials

SUMMATIVE ASSESSMENT (% of credit)
 Coursework Written Exams 20 80
DETAILS OF SUMMATIVE ASSESSMENT
Form of Assessment % of Credit Size of Assessment (e.g. duration/length) ILOs Assessed Feedback Method
Written exam – closed book 80 2 hours All Specific comments by markers and general comments on website
Coursework – based on questions submitted for formative assessment 20 66 hours overall All Specific comments by markers and general comments on website

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
All above Written exam (100%) All August Ref/Def period

RE-ASSESSMENT NOTES

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.

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

ELE – http://vle.exeter.ac.uk

Reading list for this module:

Type Author Title Edition Publisher Year ISBN Search
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]
CREDIT VALUE ECTS VALUE 15 7.5
PRE-REQUISITE MODULES ECM1702
NQF LEVEL (FHEQ) AVAILABLE AS DISTANCE LEARNING 5 No Friday 09 January 2015 Monday 12 January 2015
KEY WORDS SEARCH Supremum; infimum; series; functions; limits; continuity; derivatives; integration.