August 22, 2011
I’ve always felt a little uneasy with Lim Sup and Lim Inf’s. Today while reading Rudin’s Complex book, I finally "got" it.
To find the lim sup of your sequence: take every convergent subsequence’s limit and find the sup of those limits.
This is, of course, exactly what the definition is, but it wasn’t "visual" to me until I thought about it this way. Is there something you learned early on that you weren’t quite clear with until much, much later?
July 24, 2011
I’ve been up for a while doing practice qualifying exam questions, and sometimes I hit a point where I just do whatever it is that comes to my mind first, no matter how tedious or silly it seems. This is a bad habit. I’ll show why with an example.
Here’s the question. Let be the unit circle oriented counterclockwise. Find the integral
The sophisticated reader will immediately see the solution, but humor me for a moment. I attempted to do this by Taylor expansion. The following calculations were done:
To which the binomial theorem was applied to the numerator terms to obtain:
And at this point we note that everything is going to die off when we take the integral except the coefficient of the term. Our residue (the coefficient) will be:
which can also be written (slightly more suggestively) as:
which we should recognize as the Taylor expansion of at the point . Nice! Now we note that plugging in to take the contour integral (ignoring all those terms which don’t matter) will force us to integrate
Cutely, if we think of the Greek letter as being a "p", this solution spells out "2pie".
But now, readers, let’s slow down. This is, indeed, the correct answer. But if I had just looked at the form of the integrand, I would have seen an everywhere analytic function divided by a form of . This screams Cauchy Integral Formula. Indeed, according to the CIF, we should get the solution as
which is exactly what we got before, but only took about 4 seconds to do. It’s nice to be able to check yourself by doing something two different ways, but when time isn’t on your side (like in a qualifying exam situation, for example!) then remember:
Think before you Taylor Expand.