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Semester 1

Honours Analysis (MATH10068)

Subject

Mathematics

College

SCE

Credits

20

Normal Year Taken

3

Delivery Session Year

2022/2023

Pre-requisites

Visiting students are advised to check that they have studied the material covered in the syllabus of each pre-requisite course before enrolling.

Course Summary

Core course for Honours Degrees involving Mathematics.This is a second course in real analysis and builds on ideas in the analysis portion of Fundamentals of Pure Mathematics. The course begins with sequences and series of real numbers, introducing the concept of Cauchy sequences and results for bounded sequences. Subsequently, sequences and series of functions are introduced and concepts of uniform convergence and power series are discussed. The concept of Lebesgue integral on real line is then developed. Finally, the rudiments of Fourier series are introduced.In the 'skills' section of this course we develop and start to use some of the fundamental tools of a professional mathematician that are often only glimpsed in lecture courses. Mathematicians formulate definitions (rather than just reading other people's), they make conjectures and then try and prove or disprove them.Mathematicians find their own examples to illustrate their own and other people's ideas, and they find new ways of developing the theory and new connections. We will explore and practise these activities in the context of material drawn from some of the lectures in the course and related subjects. We will practise explaining mathematics and also consider 'metacognitive skills': the ability that an experienced mathematician has to step back from a calculation or problem, to 'zoom out' and consider whether it is developing well or whether perhaps there is a flaw in the approach. A typical example is the habit of stopping and asking whether a proof one is working on is actually using all the assumptions of the theorem.

Course Description

The course has a main section and a skills section. The main section consists of online videos, at least one weekly lecture and a weekly workshop designed to augment and extend understanding the material covered in the lectures in a smaller group setting. There will be frequent reading assignments and exercise assignments which students will be expected to have completed before the lecture.The syllabus for the main part of the course is:Review of material from Fundamentals of Pure Mathematics.Sequences and series of real numbers: Cauchy sequences, Cauchy's criterion for series, absolute convergence implies convergence.Uniform convergence of sequences and series of functions, power series.The Lebesgue integral: construction of the Lebesgue integral on the real line, the class of Lebesgue integrable functions and their integrals. Basic properties. Integration of continuous functions and the fundamental theorem of calculus. Integration and convergence.Fourier series.Skills: The content will be chosen appropriate to the learning outcomes. (10h)

Assessment Information

Written Exam 70%, Coursework 30%, Practical Exam 0%

Additional Assessment Information

Coursework 30%, Examination 70%

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