Find the order of the pole and the residue of $$f(z)=\frac{\cos z}{z^2}$$ and $$g(z)=\frac{e^z-1}{z^2}$$
What I said:
$f(z)=\frac{\cos z}{z^2}$ has a pole of order $2$ at $z=0$. Then: $Res(f,0)=-\sin(0)=0$. Can a residue be $0$?
For $g(z)$: Taylor expand $\frac{e^z-1}{z^2}=1/z+1/2!+z/3!+z^2/4!...$ which implies that $1$ is the residue. What would be the order of this?