Module: Introduction to Analysis

Module Specifications.

Current Academic Year 2024 - 2025

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Date posted: September 2024

Module Title	Introduction to Analysis
Module Code	MS323 (ITS) / MTH1054 (Banner)
Faculty	Science & Health	School	Mathematical Sciences
Module Co-ordinator	Martin Friesen
Module Teachers	-
NFQ level	8	Credit Rating	5
Pre-requisite	Not Available
Co-requisite	Not Available
Compatibles	Not Available
Incompatibles	Not Available

Repeat examination

Description

This module introduces students to the formal and rigorous approach to mathematics which underpins mathematical analysis. The students will develop the skills necessary to make the transition from a formulaic understanding of mathematics to constructing their own formal mathematical arguments, and to promote advanced mathematical thinking through the use of guided inquiry and example generation.

Learning Outcomes

1. interpret the formal mathematical definitions and statements which arise in analysis.
2. classify and describe the main components of the definitions or statements, and the motivation behind them.
3. give examples or counterexamples of important phenomena which are studied in mathematical analysis.
4. critique and explain the logical steps which are required to apply definitions or theorems to the phenomena which occur in mathematical analysis.
5. critique and explain the main logical arguments which occur in the proofs of a selection of theorems.
6. calculate important quantities which arise in mathematical analysis e.g. bounds of sets or sequences, convergence of sequences or series.

*Type*	*Hours*	*Description*
Workload	Full-time hours per semester
Tutorial	12	Discussion and feedback on weekly assignements
Lecture	24	Weekly lecture
Assignment Completion	22	Portfolio assignment
Independent Study	67	Recommended time for self-study
Total Workload: 125

All module information is indicative and subject to change. For further information,students are advised to refer to the University's Marks and Standards and Programme Specific Regulations at: http://www.dcu.ie/registry/examinations/index.shtml

Indicative Content and Learning Activities

Real numbers
Axiomatic definition of the reals, Inequalities, modulus function, triangle inequality, bounded sets, supremum and infimum

Sequences
bounded sequences, monotone sequences, convergent sequences, Cauchy sequences, Convergence theorems

Applications of sequences
Newton approximation for square roots, Exponential function and the logarithm, (Iterated fractions)

Application to Differential Calculus
Formulation of sequential continuity, Derivative, and Riemann integral in terms of sequences

Series
Geometric series, Telescoping series, Harmonic series, Leibnitz criterion, absolute convergence, Cauchy product, ratio test, root test

Power series
Convergence radius, Exponential function, Trigonometric functions, Differentiability, Integrability

Assessment Breakdown
Continuous Assessment	20%	Examination Weight	80%

Type	Description	% of total	Assessment Date
Course Work Breakdown
Assignment	n/a	20%	As required

Reassessment Requirement Type
Resit arrangements are explained by the following categories: Resit category 1: A resit is available for both* components of the module. Resit category 2: No resit is available for a 100% continuous assessment module. Resit category 3: No resit is available for the continuous assessment component where there is a continuous assessment and examination element.
* ‘Both’ is used in the context of the module having a Continuous Assessment/Examination split; where the module is 100% continuous assessment, there will also be a resit of the assessment
This module is category 3

Indicative Reading List

David Alexander Brannan: 0, A first course in mathematical analysis, Cambridge University Press, 978-0521684248
J. and P. Mikusinski: 0, An introduction to analysis, Wiley,
C. H. Edwards: 0, The historical development of the calculus, Springer-Verlag,

Other Resources

None