M337 Complex Variables TMA03 Away

Well posted the third TMA for M337 Complex Variables and just like the fifth TMA for M208 there were bits in the last question which had me stumped. Anyway a brief resume

1) Questions on Calculus of residues to solve Integrals and series
     Relatively straightforward and a joy to do

2) Questions on zero's, maxima of functions and Inverse Taylor series

     Again tricky but quite straightforward

3) a) A question showing that two series are direct analytic continuations of each other
        What you have to do is show that if the two series have a common region and they take the same values on the region then the series with the larger region can be seen as a diect analytic continuation of each other. However whilst I could see that they had a common region pages and pages of scribbling failed to convince me that the two series were equal on this region.

b) A straightforward application of the residue theorem to solve an Integral

c) The use of Wierstrass's theorem to show that a series is convergent
    Again the method is to a) show that the series is bounded by a sequence of positive terms
                                       b) show that the sequence of positive terms is convergent
   a) was straightforward but I couldn't show that the series of positive terms was convergent it failed as far as
I could see the ratio test and the comparison test.

d) Some straightforward questions on the Gamma function

So should be close to getting full marks for 1 and 2 and about half the avaiable marks for question 3. As there are only three questions I should just scrape a grade two pass.

So I'll give myself a week off to concentrate on
                     a) Cambridge Computing projects
                     b) The variational method in General relativity and it's application to the Robertson
                         Walker metric
                     c) Cubics, quartics and quintics
    I;ve found (yet another) book on Galois theory which takes a nice historical perspective and doesn't get to bogged down in formal detail until the end

http://www.galois-theorie.de/galois-theory.htm

I would probably reccomend this book as the best starting point.

Finally so far have got 8 recruits to the shared activity forum on the Cambridge Computing projects forum