Lim Sup: Life Lessons.

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?

Why thinking ahead is important: a complex integral.

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.