## Sunday, 20 March 2011

### M337 progress slow but steady

This blog topic covers my progress on  Complex Analysis, I'm about 3/4 a way through the first TMA

The questions cover the following topics

1) First part Warm up on Complex numbers and nth roots of unity
Sketch of sets and some basic inequalities Initially confusing especially as it involved the backwards form
of the triangle Inequality which is

|a+b| >= ||a|-|b||
It wasn't obvious at first what to do but I think I got there in the end

2) Transformations of a complex functions analogous to similar questions in M208
Inverse functions and their domains. I Got really bogged down here. There is an important distinction to be made between a function specified over all the domain of Complex Numbers and a restriction to a specific part of the the domain. For example take the function cosh(z) where z = a+ib because cos(iz) = cosh(z) there is a periodicity associated with cosh(z) that isn't present in the corresponding real function. As a consequence cosh(z) only has a well defined inverse if the domain of cosh(z) is restricted. Anyway my confusion arose as I failed to distinguish between a function and it's restriction.

3) Questions on Limits of Complex Functions and specification of regions. There are some intriguing links with topological concepts in this unit.

4) Differentiability of Complex Functions At last something I recognise namely the Cauchy Reimann Conditions. In most mathematical methods courses on Complex Variables these are introduced almost
immediately. This course has take 4 units to get here. However here again there is a lot of stuff being covered here about the domain of complex functions. Yes you can eventually use the same formula for differentiating real functions as  for complex variables but there are a lot of subtleties which I confess to not fully understanding yet may be second time round.

So so far this course is really stretching me I was in despair a couple of weeks ago as nothing seemed to gell but slowly I'm getting there. I suspect that as far as this course goes I will not be really forging ahead or have time to explore parallel paths. Still having got out my initial quagmire I can get on with  other courses, I hope to crack TMA02 for M208 next week

Have fun