Such a great sidenote
Mar. 30th, 2006 01:11 pmSo Tuesday was the first day of Basic Complex Variables with Narasimhan. He's known for being fast. Indeed - he started out with what complex numbers are, and then quickly moved on to differentiability, Cauchy-Riemann, the chain rule, power series, the radius of convergence, and the formal derivative has the same radius of convergence as the original series. Yay. (He started off today with the actual derivative of a power series is the formal derivative.)
So on Tuesday, as he was talking about radius of convergence, he pointed out that on the radius of convergence, the power series can converge, or not, or converge on some weird subset. But, he says, can we find for *every* subset of the circle a power series that converges precisely on that subset? No, because there are more subsets of the circle than there are power series.
Today (after proving power series differentiate as we expect), he started on the exponential function. Now, of course, the big thing is always proving that it has a nonzero period. In Dr. Nevard's class, we did this by a bunch of IVT with approximations for sin and cos. Narasimhan did it by pointing out that as exp is a homomorphism from C to C*, if we can show it's surjective, then its kernel must be nontrivial, as otherwise it would be an isomorphism, and C* has an element of order 2 whereas C does not (or, he continued, for those that haven't done group theory, there would be a z st e^z=-1, but then z≠0, 2z≠0, e^(2z)=1) and then proceeded to show that the exponential function is, in fact, surjective by constructing a power-series inverse in B1(1) and then, for the other complex numbers, taking successive square roots until it was in that region. And (after showing that any period must be purely imaginary, though of course this was all done before the surjectivity proof), why must we have a *smallest* positive-imaginary period? Because the kernel is closed, of course, and we know that e^ix≠0 for 0<x<√6 (that required looking at the series for sin, of course, but...).
The lower lounge door has been replaced, although there's no wire in the glass this time, so the next person to break it will shatter the glass and cut himself. Also, the other lower lounge door has been fixed once again. Are they going to have to fix it every few weeks?
For that matter, why doesn't the dining hall serve breadsticks anymore? Did they realize that people actually liked them? :-/
-Sniffnoy
So on Tuesday, as he was talking about radius of convergence, he pointed out that on the radius of convergence, the power series can converge, or not, or converge on some weird subset. But, he says, can we find for *every* subset of the circle a power series that converges precisely on that subset? No, because there are more subsets of the circle than there are power series.
Today (after proving power series differentiate as we expect), he started on the exponential function. Now, of course, the big thing is always proving that it has a nonzero period. In Dr. Nevard's class, we did this by a bunch of IVT with approximations for sin and cos. Narasimhan did it by pointing out that as exp is a homomorphism from C to C*, if we can show it's surjective, then its kernel must be nontrivial, as otherwise it would be an isomorphism, and C* has an element of order 2 whereas C does not (or, he continued, for those that haven't done group theory, there would be a z st e^z=-1, but then z≠0, 2z≠0, e^(2z)=1) and then proceeded to show that the exponential function is, in fact, surjective by constructing a power-series inverse in B1(1) and then, for the other complex numbers, taking successive square roots until it was in that region. And (after showing that any period must be purely imaginary, though of course this was all done before the surjectivity proof), why must we have a *smallest* positive-imaginary period? Because the kernel is closed, of course, and we know that e^ix≠0 for 0<x<√6 (that required looking at the series for sin, of course, but...).
The lower lounge door has been replaced, although there's no wire in the glass this time, so the next person to break it will shatter the glass and cut himself. Also, the other lower lounge door has been fixed once again. Are they going to have to fix it every few weeks?
For that matter, why doesn't the dining hall serve breadsticks anymore? Did they realize that people actually liked them? :-/
-Sniffnoy
