Saturday, April 17, 2010

Excursions into the infinite complex plane.

So much has happened since I last wrote ... I'm still working on making sense out of it all.

Homework has been keeping me busy, as usual. I once again have several things due over the weekend, so I will be busy working on that. Between electromagnetism, partial differential equations, modern physics, and calculus IV, I have plenty to keep me occupied.
As if that weren't enough, I got my computer back from the campus tech guy, who was unable to find anything wrong with it other than the fact that the computer simply refuses to acknowledge the wireless module, and it still locks up, with no pattern that I can seem to figure out, yet in an oddly non-random way. I'm unsure which way to go at this point ... I hardly have time to deal with warranties and such, but I could really use my computer working fully as the end of the semester approaches.

Before I forget, I'd like to say a big 'thank you' to my aunt and uncle who were out here this week for everything -- the dinners were great, and the conversation and catching up were awesome. It was really nice to take a break and focus on something other than school for a while. So, mahalo nui loa! (thank you very much)

 In other news, I managed to solve another problem for my Complex Analysis professor today, which was the question of the convergence or lack thereof of the geometric series in the complex plane on the unit circle. If you don't know what I'm talking about, don't worry, you're not missing much. Infinite series and sequences are some of my least favorite subjects from Calculus II, and from the people I've talked to, many others feel exactly the same way. They inspire...strong feelings, to put it mildly. This is because they are very complicated, and most of the time you work with them is spent trying to figure out if a particular series converges or not. (Convergence means that if you add up an infinite number of terms from the series, you will get a finite number. Divergence simply means that you get infinity. This can be a surprisingly difficult thing to do) The geometric series is one of the few series for which you can actually find what it converges to; its terms look like \(x^n\), for n from zero to infinity. (Its first few terms are \(1 + x + x^2 + x^3 + ...\)) It is well known that for any x less than 1, the series  converges, while for any x greater than 1 it diverges. This is because its radius of convergence is equal to 1, and it turns out that every infinite series has such a radius, inside which it converges and outside which it diverges. On the real number line, the word 'radius' doesn't seem to make much sense, because there are only two points for the boundary, but in the complex plane it makes perfect sense because you are dealing with a plane in which numbers can approach from any direction, instead of merely from the left or right.
Anyway, in the real case, it can be shown that the geometric series diverges at \(x = 1\) and \(x =-1\), for different reasons. But that's only two points. The question in the complex case is, what happens for all the other infinite number of points on the circle of radius 1 around the origin? (written symbolically as \(|z|=1\)) However, there is a rather simple (once you see it) and elegant proof that the geometric series diverges for all \(|z|=1\), and I was able to be the inspiration for my professor that enabled him to discover it today (though as usual, the proof he came up with was much more elegant and simple than the clumsy one I had cobbled together). 

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