You do not want to find the zeroes instead you want to find the functions singularities.
This is when cosz=0.
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
(z)/((z^2+1)(9-z^2)
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.
Any help?
It doesn't mean it must converge 'only' on the circle it is just that the domain includes the circle.
The second one is easier. It has singularities at i, -i, 3 and -3. It asks for convergence at 2i but the series we want will converge at 1<|z|<3 which include 2i.
So I got this expression writing the things out in partial fractions
z/20 * ((-1/(1-iz) + (1+(2-iz)+(2-iz)^2+...) + 1/(1-(4+3z)) - (1/9)*(1+(z/3)+(z/3)^2+...))
Where do I go from here? The two terms which I wrote out in geometric series form don't converge for 1<|z|<3