## Monday, 13 June 2011

### Back on Track M337 TMA02

After a fairly intensive weekend managed to get back on track with Complex Variables and have just posted off the 2nd TMA

The topics covered were

1) Complex integration
2) Cauchy's theorem
3) Taylor series
4) Laurent series

These form the hard core of any Complex Variable theory course. The questions covered the following

1) Simple integration along two different paths of a function showing that the two integrals were not the same
and hence that the function was not entire. Then a couple of complex integrals which can be treated in a similar manner to ordinary integrals

2) 1st part a relatively straightforward application of Cauchy's Integral and derivative formula to integrate a function

2nd Part (Probably the hardest) a question to really stretch your understanding of Liouville's theorem which states that a function which is entire over the whole of the complex plane and bounded must be constant, Some parts had me really scratching my head here but I think I got there in the end (time will tell).

3) Exercises on Taylor series, not much different from stuff in MS221 but the conditions for convergence etc involve the |z|
Then another tricky question on the uniqueness theorem which states that if two functions can be shown to identical at a limit point on a region which is a subset of C then they are identical all over C. I think this is a warm up exercise for the topic of analytical continuation

4) Exercises on Laurent series and an application of the Residue formula

PArt of the problem is learning how to phrase the proofs and assumptions and that can only come with practice. I would love to be able to write phrases such as

"This function is analytic apart from the points a1, b1 and c1 where it has poles of order 2 and 3 at b1 and c1 and an essential singularity at a1, Thus on the contour C1 we can integrate the function but not on C2 ..." naturally, before diving into the calculation  but that will not happen immediately.

So how do I feel not sinking but missing out huge chunks of theory and it's not coming naturally just yet, but slowly but surely it's sinking in. So my predictions for the course a grade 2 is feasible but I wouldn't feel confident about a distinction

The pressure is not really off yet as I have just over a  week to do my next TMA for M208 as this is real analysis it will be interesting to see if all that work I did reading Brannan in between MS221 and M208 has paid off. I have to say that whilst analysis is interesting it doesn't grab me in the way that group theory does,

"Analysis is for the head but group theory is for the heart"

Looking forward to getting to the 2nd part of group theory for M208.

1. Chris,
I know this is a "how long is a piece of string" question but how does the level of difficulty with complex analysis compare with M208's real analysis?
M337 is on my radar for October 2012 - I enjoyed the M208 analysis and found it a fair bit easier than the linear algebra within the course.
Any thoughts?
Thanks.

2. Hi Chris,
I fully agree - complex analysis is a must for the maths war-chest.
It's all my mate's son, who's off doing a PhD at some posh London uni, ever mentions - seems to hold the answer to the universe somehow. :-)
As I plan to take the course, I'm dismayed, nay, gutted, to see you use the words epsilon delta.
Out of everything on M208 last year, this concept turned out to be the only thing I ended up with a poooooooooooooooooooor knowledge of. I've no idea why as the notion doesn't seem all that complicated - but it had a knack of confuddling my grey matter every time. My usual trick of Youtubing the heck out of it failed to help much.
As for my blog on MST209 - my over-riding impression is that it's significantly harder than M208. But then, having no physics background at all, everything from the notion of a normal force to a damping constant has been utterly new to me as well - compounding my confusion.
Good luck on your next batch of assignments - I have no idea how you do it.

3. I'm afraid epsilon delta is a key part of any analysis course, M208 can lull people into a false sense of security in that it is possible to bypass it and also skip the proofs but still get good marks in assignments and exams. It really is worth putting the effort in to try and understand it.

As for MST209 if you don't have a physics background or didn't do Applied maths at A level I can see that a lot of it would be confusing. Still it lays the foundations for mathematical modellig so worth sticking at. Having looked at some of the past papers for interest I can see that the problems would be tricky. Also the workload seems far greater than say M208.
One thing I would find off putting is the amount of multiple choice I really don't see that anything is gained by this especially for a maths course. If an answer is close you get penalised unfairly compared to say someone who just guessed the answer and maybe got it right.
Glad I don't have to do anymore.

4. I'll be continually epsilon deltaing myself once MST209 is out of the way. :-)
The multiple choice you refer to is on the first section of the exam only and I, too, am not a fan.
The TMAs are, of course, devoid of multiple choice questions.
I'll continue to check your posts regularly - your blog is a valuable present tense gaze into my future course/s.
So, thanks for that. :-)

5. Your welcome I find your blog interesting as well Multimode oscillations is a really clever and neat synthesis of linear algebra and mechanics it also forms a key part in quantum mechanics. There is a reason for all that stuff on oscillations the basic maths turns up over and over again in all sorts of physics problems I'm quite envious really.