Complex Analysis: Laurent Series
I'm having some trouble with these two Laurent Series questions:
1) Consider the Laurent Series expansion of (pi^2-4z^2)/cos(z) which converges on the circle |z|=5. What is the principal part of this expression. What is the largest open set on which the series converges.
So I have no idea how to begin, especially with the series having to converge on the circle |z|=5. I know we can rewrite the expression as
(pi-2z)(pi+2z)/cosz. so we have zeros at +/- pi/2. Past that I am lost.
The second question is similar:
2) Find the Laurent Series expansion of the function
centered at 0 and convergent at z=2i. What is the largest open set for which this series converges.
For this one I am confused because I thought we used the Laurent series expansion to fix problems on an annulus.