## MAS221 Analysis 2016-2017## Semester 2## Lecturer: Prof Sarah WhitehouseImage source: Riemannsches Integral – Wikipedia
## Semester 2 Course Information
A good grasp of Semester 1 material is essential to understanding the module in Semester 2. This semester we will continue the study of analysis of functions of one variable, building on what you saw in Semester 1. Later on we will generalise to the study of functions of several variables. Here is a summary of the material we will cover in semester 2, with a rough indication of how many lectures will be devoted to each topic. - Convergence of series in terms of the sequence of partial sums. Examples, including geometric series and harmonic series. Tests for convergence: comparison test, absolute convergence, ratio test. (3 lectures)
- Definition of the Riemann integral. Basic properties. Examples. Continuous functions are Riemann integrable. The fundamental theorem of calculus. Improper integrals. (3 lectures)
- Sequences and series of functions; pointwise and uniform convergence. Proof that a uniform limit of a sequence of continuous functions is continuous. Example to show the same is not true of pointwise limits. Uniform continuity. The uniform convergence theorem for integrals, and application to differentiation. Series of functions and the Weierstrass $M$-test for uniform convergence. Example of an everywhere continuous but nowhere differentiable function. (5 lectures)
- Applications I; power series, radius of convergence, termwise differentiation. Exponential function and e. (2 lectures)
- Sequences in R^k. Distances in R^k. Convergence of sequences in R^k. Open balls and open sets. Intersection and unions. Examples. Cauchy sequences revisited. (3 lectures)
- Functions of several variables. Images and pre-images, continuity, algebra of continuous functions. Examples, including polynomials. Uniform continuity. Examples. (3 lectures)
- Applications II. Differentiation under the integral sign. Applications to the B and Gamma functions. (2 lectures)
## TestThere was an in-class test in the Thursday lecture of week 1.- Test questions
- Test solutions (An annotated version with commentary will appear once I have finished marking the tests.)
Important comment: The test questions are comparable to the "easy marks" part of the exam; you need to be getting pretty much all of these marks to pass the exam. ## Lecture notesHere are the lecture notes for the whole of the semester.This version contains the statements of definitions and results, but not proofs or solutions to examples. You are expected to attend the lectures and take your own notes of the proofs and the solutions to examples. (I expect to supply some version of this material in typed form eventually, but this is unlikely to contain everything that I explain in the lectures. In particular, pictures and graphs are important to aid understanding of the material. I expect to draw lots of these in lectures, but not to produce a printed version of them.) ## ExercisesHere is the complete set of exercises for the semester.Problem classes These take place on Tuesdays (at 11 or 2) in weeks 2, 4, 6, 8, 10 and 12. Week 2 class: we'll focus on the following problems: Chapter 1, Qs 1, 3, 4, 5, 6. Week 4 class: we will go through Q2(b) from the test and then focus on Chapter 2, Qs 5, 3 4(a), (b), (d), 6.
Homework 5 from semester 1 was Qs 83, 84, 88, 91 and 98. This is due in at the week 2 tutorials. Homework 1 for this semester is Chapter 1, Qs 7, 8 and 9 and Chapter 2, Question 2. This is due in at the week 4 tutorials. Homework 2 for this semester is Chapter 2, Qs 7 and 12 and Chapter 3, Q2. This is due in at the week 6 tutorials. Solutions Solutions will appear here as the module progresses.Solutions to Chapter 0 Problems: Revision ## Past Exam Papers
The last two years' exam papers are available from here. The format of this year's paper will be the same as the 2015-2016 paper. ## Contact Details
Prof Sarah
Whitehouse
You
can always make an appointment by email if you cannot come to an office
hour. Last updated: 24
February 2016 |