Value of integral over $|z+1|=1$.

Question

Evaluate $\displaystyle\oint \frac{z^{2}}{z^{4}-1} dz$. How to know the points lie inside the curve or outside the curve?

Answer 1

Substitute $z+1=t$ $$\implies \displaystyle\oint \frac{(t-1)^{2}}{(t-1)^{4}-1} dt $$

If $\alpha$ is one of the roots of the denominator,then if $$|\alpha|-1>0\implies \alpha$$ lies outside the region or in other words, contour integral is zero. Else if,

$$|\alpha|-1<0\implies \alpha$$ lies inside the region