Find the Laurent Series representations in powers of $z$ for
i) $\frac{ \cos{z}}{z}$
ii) $z^4 \cosh{\frac{1}{z^2}}$
Where do they converge?
I found the Laurent Series for each of the functions, yielding the result
i) $\sum_{n=0}^\infty\frac{(-1)^nz^{2n-1}}{(2n)!}$
ii) $\sum_{n=0}^\infty \frac{z^4}{(2n)!(z^2)^{2n}}$
Both of these functions have a pole at $z=0$, so I want to say that the answer to where they converge is $z \neq 0$. However, I am under the impression that a Laurent series should have a convergence along the lines of $r< |z-z_0| < R$.
Just take $z_0:=0$ giving $$r< |z-0| < R$$ or $$r< |z| < R$$ for the desired set of convergence.