I have not checked your work but here is something that may help with the second part of the question. Once on the curve , you have:
I'm stuck on a question where I'm meant to use Cauchy's integral formula. Here's the question:
by Cauchy's integral formula. Computing this integral along the path show that
This is how I approached it:
Let . has poles at and .
The path is a circle centered around with a radius of 1. Consequently, is outside of as . Therefore, according to Cauchy's integral theorem,
where is an arbitrarily small circle around . (I'm not sure if this is right, but I'll show the rest of my work assuming it is.)
Around , is analytic, with
Then, using Cauchy's integral formula,
I've now got no idea how to relate that to the second integral in the question, or if it's even right. Any help is very much appreciated
In such a case the best is to set so that the integral is...
... and the path is the 'unit circle' for which is with . If the only pole inside the path is at so that is...
... so that the result found by 'Zeus' is exact! ...
Now let suppose that we don't remember the Cauchy's formulas, so that we have to find the integral (1) in 'standard way'. In that case we can set so that the integral becomes...
Now if we confront the (2) and (3) it is almost immediate find that is...
It seems to be all right... but 'Zeus' wrote in his initial post...
A little amletic doubt! ...