## Why thinking ahead is important: a complex integral.

### 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.**

I actually did it in my head by factoring out an e, leaving me with e*e^(z^2), and then Taylor expanding. This avoids the complications that the binomial expansion brings.