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.