Introductory description

At the beginning of the nineteenth century the familiar tools of calculus, differentiation and integration, began to run into problems. Mathematicians were unsure of how to apply these tools to sums of infinitely many functions. The origins of Analysis lie in their attempt to formalize the ideas of calculus purely in the language of arithmetic and to resolve these problems.

Module web page

Module aims

Outline syllabus

This is an indicative module outline only to give an indication of the sort of topics that may be covered. Actual sessions held may differ.

You will study ideas of the mathematicians Cauchy, Dirichlet, Weierstrass, Bolzano, D'Alembert, Riemann and others, concerning sequences and series in term one, continuity and differentiability in term two and integration in term one of your second year.
By the end of the year you will be able to answer many interesting questions: What do we mean by `infinity'? How can you accurately compute the value of π or e or √2? How can you add up infinitely many numbers, or infinitely many functions? Can all functions be approximated by polynomials?

Learning outcomes

By the end of the module, students should be able to:

• Understand what is meant by `infinity'
• Develop their own methods for solving problems
• Accurately compute the value of π or e or √2
• Add up infinitely many numbers, or infinitely many functions
• Answer: Can all functions be approximated by polynomials?

Indicative reading list

M. Hart, Guide to Analysis, Macmillan. (A good traditional text with theory and many exercises.)
M. Spivak, Calculus, Benjamin.
R.G Bartle and D.R Sherbert, Introduction to Real Analysis (4th Edition), Wiley (2011)
L. Alcock, How to think about Analysis, Oxford University Press (2014)

Subject specific skills

Analysis gives first-year undergraduates a first excursion in to pure mathematics. The students will gain a new perspective and a deeper understanding of familiar mathematics which they have seen in school (e.g. real numbers, functions and differentiation). In Analysis, these concepts are developed with mathematical rigour, which characterises much of university mathematics to follow.

Transferable skills

Critical thinking
Group work (through group supervision)