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.
You will study ideas of the mathematicians Cauchy, Dirichlet, Weierstrass, Bolzano, D'Alembert, Riemann and others. Following on from the material concerning sequences, series and continuity in term one, you will study differentiation and integration in term two.
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 integrate a function?
There will be considerable emphasis throughout the module on the need to argue with much greater precision and care than you had to at school. With the support of your fellow students, lecturers and other helpers, you will be encouraged to move on from the situation where the teacher shows you how to solve each kind of problem, to the point where you can develop your own methods for solving problems. You will also be expected to question the concepts underlying your solutions, and understand why a particular method is meaningful and another not so. In other words, your mathematical focus should shift from problem solving methods to concepts and clarity of thought.
M. Hart, Guide to Analysis, Macmillan. (A good traditional text with theory and many exercises)
K.G. Binmore, Mathematical Analysis: A Straightforward Approach, CUP (1982)
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)