I didn’t have the best day Thursday; I was very sick (felt as if I had been in a boxing match..chills, aches, etc.) but was good to go on Friday (no cough, etc.)

So I walk into my complex variables class seriously under prepared for the lesson but decide to tackle the integral

Of course, you know the easy way to do this, right?

and evaluate the latter integral as follows:

(this follows from restricting to the unit circle and setting and then obtaining a rational function of which has isolated poles inside (and off of) the unit circle and then using the residue theorem to evaluate.

So And then the integral is transformed to:

Now the denominator factors: which means but only the roots lie inside the unit circle.

Let

Write:

Now calculate: and

Adding we get so by Cauchy’s theorem

Ok…that is fine as far as it goes and correct. But what stumped me: suppose I did not evaluate and divide by two but instead just went with:

$latex where is the upper half of ? Well, has a primitive away from those poles so isn’t this just , right?

So why not just integrate along the x-axis to obtain because the integrand is an odd function?

This drove me crazy. Until I realized…the poles….were…on…the…real…axis. ….my goodness, how stupid could I possibly be???

To the student who might not have followed my point: let be the upper half of the circle taken in the standard direction and if you do this property (hint: set . Now attempt to integrate from 1 to -1 along the real axis. What goes wrong? What goes wrong is exactly what I missed in the above example.