Saturday, March 20, 2010

The Residue Theorem

Today was, all in all, a rather interesting day.

To understand why, I need to back up to Thursday night, when I was finishing up my homework for Complex Analysis. Of the ten problems assigned, nine of them took my twenty minutes or less to complete (because we'd gone over how to work them in class already). The last one was a stumper. In principle, it was a piece of cake after the problems I'd been doing: simply perform an integration over a square region over four different complex functions. Yet over the next three hours I attacked that problem with every trick, technique, or theorem I know of in both the real and complex plane, to no avail. I finally gave up frustrated at 10 o`clock, because I had been up late for the last few nights and really couldn't stay awake any longer, with the intention of working on it in the morning before class.

Before retiring for the night, though, I hit upon a method to compute the integral, in principle, assuming several conditions were met. So this morning I began chipping away at the problem again. After removing all the difficulties to my satisfaction, I proceeding to the tedious (for me) work of actually writing out the problem, made all the more tedious because I anticipated several pages of computations.

And that's when it hit me: after doing a bit of preliminary setup, the closed nature of the path of integration meant that most of the results canceled in pairs, leaving only a simple complex integral, which I did in about half a minute. You can hardly imagine my elation when I realized that I was only dealing with a page or two of writing for all four integrals, instead of the the two pages each I had been envisioning.

Even with the reduced amount of writing, I still wasn't able to finish it before class, but the first thing Dr. Figueroa did was to announce that the homework was postponed until after Spring Break, because he hadn't been able to work out that very same problem with the material we've covered so far, despite staying up till one in the morning working on it (we're using a free textbook from off the Internet, which doesn't come with solutions). He mentioned that the problem could be solved using techniques later on in the book, using something called 'the method of residues' which caught my ear because it sounded like an apt description of what I had found: most of the problem disappeared, leaving only a 'residue', a single, simple, complex integral. When I mentioned I had figured out how to solve the problem, he was intrigued, and had me demonstrate it at the white-board. When he figured out where I was going, he got quite excited, jumped up from his chair, and immediately proceeded to put it all on a firm theoretical footing, then discovered a way to reduce it from the clumsy sum of four parameterizations I had been using to an elegant single parameterization over a circle, which reduced the problem down to a few lines. I think I made his day; he was still pleased as punch when class ended.

I did a little digging, and apparently what I discovered (arithmetic errors aside; I didn't have time to fully develop it) is know as the Residue Theorem, and is a standard and well-known tool in complex analysis.

The other interesting part of my day was the part where I and two classmates attempted to build a working cloud chamber for our project for my Modern Physics class and the fun I had with some of the leftover dry ice, but it's getting late, and that actually has some pictures to go with it, so I will save it for tomorrow when I can really do it justice.

3 comments:

  1. Ah yes, the Residue Theorem is sometimes referred to as the "Crown Jewel of Complex Analysis" because it allows you to compute a lot of real integrals that would otherwise be impossible without numerical methods.

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  2. And after using it on my homework, I can see why it's called that!

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  3. Ah, so *that* was the problem he couldn't solve....I wish I knew more about how it actually worked!

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