Help Needed - Region of Convergence for Laurent Series of f(z)

[Butelle]

Hello -

I have a problem in general finding the region in which the Laurent series converges...

Could someone please help me with this question - I know that this is is meant to be easy (as there is no fully worked solution to this) but I don't understand it:

f(z) = 1/ [(z^3)*cosz]. The function has a Laurent series about z = 0 converging at Pi/4. What is the region in which this Laurent series converges?

Thanks.

 

[jasonRF]

When your textbook derived the Laurent expansion, what was the shape of the domain that the function had to be analytic in (it is complex analysis after all - isn't everything about being analytic?). What can cause a function to not have that property, and hence will make the series not converge?

 

[Butelle]

I'm sorry I am not really understanding? I derived the expansion and then I didn't know what to do after that?

 

[HallsofIvy]

He is saying that the series will converge until it hits a point at which the function is NOT analytic. This particular function, f(z)= 1/(z³cos(z)), is analytic except where the denominator is 0: z= 0 or z an odd multiple of \pi/2. That is, the Laurent series around z= 0 converges for 0< |z|< \pi/2.

 

[Butelle]

ahhh ok i understand - see i didnt even understand the theory behind it! i get it now - thanks! :)