Ahh, today is a day of great rejoicings! I have finished and turned in the last homework assignment for this semester (true, I have a take-home test for Partial Differential Equations due on Monday, but that is much preferable to an in-class test in that class). I was highly honored today by Dr. Figueroa asking for copies of my last two homework sets to use as the basis for the answer key he was compiling for them, since I typically get most of the problems right (I may have mentioned that the free textbook we're using has no answers, so he has to compile answer keys for the homework he assigns himself).
Speaking of the most recent homework we had for Complex Analysis, I would be remiss if I didn't mention the Residue Theorem as promised. The Residue Theorem is often called "the crown jewel of complex analysis", and now I know why. Problems that previously would have been incredibly difficult, or even impossible, are now a snap. I didn't get a chance to work on the homework until Monday, and expected to be spending upwards of ten hours on it, and was very pleasantly surprised as problem after problem fell to the power of the Residue Theorem. I finished approximately 90% of the problems in less than two hours, and took only about another hour for the remaining ones. The feeling was one of delight, as previously-impossible problems became simple ones. I can hardly wait to actually use it in a physics application now.
The details are a bit deep (it took a whole semester to build up to them!), but basically, it says that when integrating over a region that contains one or more singularities (places where division by zero occurs), the integral is equal to 2πi times the sum of the residues, which are the values of the function being integrated attains at the singularities. Since plugging in the number directly would be undefined, residues have to be found another way, which can be as easy as a single differentiation (though they can certainly be difficult to compute, as well). The beauty of the theorem, is that it completely removes the need to actually integrate anything, which is very nice (differentiation is a science; integration is an art [integration is the inverse of differentiation, if you haven't taken calculus, and while you learn differentiation in first-semester calculus, it takes the rest of that semester plus another three to really get a handle on integration]).
Tomorrow night, I have been invited by Dr. Takamiya to go on an observing run. We won't be going up Mauna Kea; instead, we'll be controlling the telescope from the Institute for Astronomy building near campus (I'm not sure which telescope it is, though. I'll let you know when I find out). I also don't know how long I'll be there, it could be an all-nighter! I'll be sure to let you know all about it Friday.