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.
then it becomes:
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?
I worked on it a bit more. Please tell me if I am on the right track.
Using partial fraction I constructed 2 circles around z=1 and -1.
where is the circle around z=1 and is a circle around z=-1
Letting and using the Cauchy's Integral Formula
Here, f(z) must be analytical because all the partial derivatives are indeed 0. Is this right?
Is this correct?