Thread: 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

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?

Classifications =removable, essential