At $z=0$, the function $$f(z)=\exp\left(\frac{z}{1-\cos(z)}\right)$$ has infinitely many positive and negative powers of $z$.
I want to determine whether it is True or False.
Taking $\lim_{z\to 0} f(z)$, we note that limit does not exist so the singularity at $z=0$ is essential which means there are infinitely many negative powers of $z$ in the Laurent series expansion of $z$.
But I am not able to see why it should have positive powers.
I tried looking at series expansion of $f(z)$ which turned out to be
$$1+{z\over 1-\cos(z)}+ ({z\over 1-\cos(z)})^2{1\over 2!}+\dots$$
but I am not able to properly argue about why infinitely many positive powers of $z$ should be present.