# Find the poles of f(z)=11+zwf(z)=\frac 1{1+z^w} for w>1w \gt 1

I am trying to use contour integration on the following integrand between $0$ and $\infty$, however I am not sure how to go about finding the poles for it:

Consider the denominator equal to zero:

How would I go about determining the types of poles we would have for $f(z)$ given the many different forms it could take dependent on $w$?

so $w$ is a natural number greater than $1$. So all you need is to solve $z^w = -1$, that is to say, all the $w$-roots of $-1$.
These are unique and there are exactly $w$ of them, so your function will have poles of order $1$ at the $w$ roots of $-1$. They all lie on the unit circle.