Archive for March, 2014

A few analysis resources

March 12, 2014

This will be my final post associated with the Analysis I course, for which the last lecture was yesterday. It’s possible that I’ll write further relevant posts in the nearish future, but it’s also possible that I won’t. This one is a short one to draw attention to other material that can be found on the web that may help you to learn the course material. It will be an incomplete list: further suggestions would be welcome in the comments below.

A good way to test your basic knowledge of (some of) the course would be to do a short multiple-choice quiz devised by Vicky Neale. If you don’t get the right answer first time for every question, then it will give you an idea of the areas of the course that need attention.

Terence Tao has also created a number of multiple-choice quizzes, some of which are relevant to the course. They can be found on this page. The quiz on continuity expects you to know the definitions of adherent points and limit points, which I did not discuss in lectures.

How do the power-series definitions of sin and cos relate to their geometrical interpretations?

March 2, 2014

I hope that most of you have either asked yourselves this question explicitly, or at least felt a vague sense of unease about how the definitions I gave in lectures, namely

\displaystyle \cos x = 1 - \frac{x^2}{2!}+\frac{x^4}{4!}-\dots


\displaystyle \sin x = x - \frac{x^3}{3!}+\frac{x^5}{5!}-\dots,

relate to things like the opposite, adjacent and hypotenuse. Using the power-series definitions, we proved several facts about trigonometric functions, such as the addition formulae, their derivatives, and the fact that they are periodic. But we didn’t quite get to the stage of proving that if x^2+y^2=1 and \theta is the angle that the line from (0,0) to (x,y) makes with the line from (0,0) to (1,0), then x=\cos\theta and y=\sin\theta. So how does one establish that? How does one even define the angle? In this post, I will give one possible answer to these questions.