I was asked to evaluate the following:
where C is the circle
Well I figure that is the function
is analytical then according to Cauchy's intergral theorem:
This is somewhat complicated since it's difficult to separate the function into the real and imaginary part to take their derivatives. I pluged it into the calculator and it seems like it's not analytical. So, I went about trying to do it.
and
then it becomes:
let:
and
then:
and
And using by parts:
since
r=2, and we have:
Now pluging in the limits
and 0 we have:
How can this be? It shouldn't be 0 since it's not analytical. What am I missing here?