For some time since 2018 my maths studies had fallen by the wayside and I was beginning to miss that regular challenge of learning something new. About this time last year I decided to pick up The Open University's M337 Complex Analysis course again and I have been plodding along with it very slowly. I started from the beginning again because I thought I would have forgotten what I had already learnt.
So, at the moment I am nearly at the end of book A3 on continuity and, if anyone wonders, this is not the advised rate of reading for study of this course at the OU (it should take a year to finish the whole course). Not that I am worried. This is just an interesting slow ramble for me and sometimes I don't pick it up again for weeks at a time. There has been nothing so far that has completely flumoxed me and it all seems relatively familiar having previously studied the OU's M208 (Pure Mathematics). I do like to work consistently through all the problems and exercises and this is probably why I tend to go so slowly - these are the best bits!
At the moment I am on chapter 5 of A3 and have enjoyed this diversion into sets and the Extreme Value Theorem and I am sure it will play its part in later books. I have just been working on Problem 5.5 and part (b) had me tying myself up in knots as I was trying to find an estimate for the upper bound of |sin z| for the set {z : |z| <= 27} without using the Triangle Inequality. This wasn't intentional - I just had forgotten to apply this theorem. In the end I won out but it was messy. I think by the time I had finished I had a 'better' (i.e. lower value) estimate than the triangle inequality gave. My estimate was that |sin z| <= (1/2)(e^54 + 3)^(1/2). The OU solution using the triangle inequality was |sin z| <= e^(27).
This reminded me that there was a similar remark about 'best possible' constraints at the end of the chapter on inequalities in book A1. It said that the inequality |z^2 - 4z - 3| <= 15 for |z| = 2 is not the 'best possible'. It said with more work it is possible to prove that the best possible inequality is |z^2 - 4z - 3| <= 7(7/3)^(1/2). This, of course, set me off trying to work this out for myself and after a couple of pages of working out I agreed. Here is the result written out in my fair hand:-
