Here is the integral I am dealing with:
$\int_{|z|=1}\frac{z^{11}}{12z^{12}-4z^9+2z^6-4z^3+1}dz$
I have been advised to make the change of variable $w=\frac{1}{z}$ which I believe results in:
$\int_{|w|=1}\frac{-1}{w^{13}-4w^{10}+2z^7-4w^4+12w}dw$
Now, Rouche's Theorem tells us the denominator (of the $w$ integral) has one root inside the unit disc (right?) so $w=0$ is the only pole we need to consider ?
Is this approach valid? Do I just need to calculate a residue now?
Are there alternative approaches that might work better, maybe not involving a change of variables?