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 , however I am not sure how to go about finding the poles for it:

f(z)=11+zw,wZ:w>1

Consider the denominator equal to zero:

1+zw=0

zw=1

z=11wwz

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

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