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

f(z)=11+zw,w∈Z:w>1

Consider the denominator equal to zero:

1+zw=0

⇒zw=−1

⇒z=−11w≡w√z

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?

**Answer**

so w is a natural number greater than 1. So all you need is to solve zw=−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.

**Attribution***Source : Link , Question Author : Jeremy Jeffrey James , Answer Author : Ant*