Sunday, October 12, 2008

If only complex logs were that simple...

Today I have woken up on the wrong side of bed.

Normally I would make a big deal about that statement for it isn't physically possible for me to do such a thing, but today you'll have to forgive me for not trying to be remotely funny.

Once again I have been a couch potatoe since I woke up, but at 2pm I forced myself off the sofa and made a cup of tea for myself. This was an achievement indeed for I then proceeded to my desk and *drum roll* some applied complex analysis.

I have to train myself to endure this subject for two hours at least, because unsurprisingly, after an hour or so of just going through the first lecture notes and recalling some information from a text book, my head is going to explode. I honestly am in need of another cup of tea now which is a bad thing! (I had reduced my tea intake to two cups a day, and now I'm heading back to three or four per day again.) But I best get used to this head exploding feeling and learn how to carry it with me without it affecting my stability, for the sake of my Topology lectures if nothing else.

We have only had four lectures on this module but we are expected to have remembered what we did in Complex Analysis last year. Now that's where the problem lies. I remember things and things about things, but nothing about the things that I need to remember! I mean Cauchy-Riemann equations are one of the things that I need to remember, but to remind myself I have to now fish my last years folders out from where ever they are. (I can remember the idea of the proof of Cauchy-Riemann equations, but strangely not much else!)

You might be happy to read, or perhaps proud (!) to see that I have matured an ickle bit from last year. Now I shan't mention what might give you such a silly idea, for if you haven't noticed it then you are not from those who I speak off. (But since I can't resist saying one thing, I will say that the above paragraph is the biggest clue in this post...).

So what about complex logs was I meant to say? Well write if you're being so and so-ish! Instead of writing anything out again, here's the beginning of what the problem is - the function ln(z):
Now my writing is awfully unlike itself there, so I am going to lie and say that I was squinting due to the pain in my head, and couldn't see much. The truth is that my time for complex analysis was nearing its end... (my tea had finished).

In applied complex analysis one idea that is quite difficult to get your head around (initially) is that of the branches and cut planes. The complex logarithm is multi-valued (the 2n\pi above) and we don't like that at all. Hence we do some cutting here and there, to find ourselves a nice range where this silly function is single-valued and "nice". (Nice could mean analytic which just means that a function is differentiable at every point inside a domain D.) Things get ugly along the branch, but we can ignore that for now and just say that our point cannot cross any cuts, for the previous "niceness" disappears. (Pictures are needed for this in my opinion, but I have a student coming soon so that's not possible now).

For ln(z), we can take the branch cut to be anywhere we want, and examples of two particular cuts are:
a) the "Principal Branch" : - \pi (arg z + 2n\pi) \leq \pi and
b) 0 \le (arg z + 2n\pi)  2\pi.

We often choose the branch in such a way that, for z real, we would have just the "normal" value of \ln z= \ln(x+iy) = \ln x.

It's when we get to functions like:

 \ln \left(\frac{z-1}{z+1}\right),
that things get more interesting. (I invite readers to guess what the branch cut would be in this instance.)

I realise that this isn't a very meaty post, and I hope to fill the details in soon (especially about the last function mentioned). It is my intention to post my lecture notes on at least one module this semester and see if it helps me in anyway. Some people go home and write their whole notes out again, (once understood), which is a very good way of keeping on top of things. Although I would dearly love to write notes on this module due to its level of difficulty, I am slightly reluctant due to having to insert images etc.

Anyway, as always please feel free to correct any nonsense that I write, but I will have to run now for I need tea before that student comes!

PS: Any faulty inequalities will be edited later - silly html won't allow the less than symbol!

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