Locate & classify singularities!!

I have spent about 2.5 hours on this question and I just can not seem to get anywhere. Please can someone show me how to do this.

**Locate and classify the singularities (giving the order of any poles) of the function**

f(z) = z^3/(1-cos z)^2

*Hint : Use cos z = cos (z - 2kpi ), for all K E Z*

My attempts so far....

This function has singularities at 0, +/- 2pi, +/-4pi, +/-6pi; (As this is where denominator=0)

Also f is analytic on each punctured disc D= {z:0 < l z-2kpi l < 2pi}

I have seen a previous example, (z/sinz) whereby a hint of sin(z-kpi)=(-1)^ksin z was used and wondered if I had to apply a similar approach?

Please help as I am experiencing difficulties in using the Laurent's theorem?

Shoni

Re: Locate & classify singularities!!

Hey shoni.

You may want to consider the case where z = 0 (if this is really a singularity or whether it converges). I have a feeling it may converge but I could be wrong.

Also, what kinds of classifications are you asked to give for the singularities?

Re: Locate & classify singularities!!

Classifications =removable, essential