I am trying to find the type of singularity the function $f(z) = \exp\left(\frac{(\cos z-1)^2}{z^4}\right).$ has at $z = 0$.
The expression reduces to $\exp\left(\frac{\sin^4(z/2)}{z^4/2}\right).$ The function has removable singularity at $z=0$ as $\lim_{z\rightarrow 0}zf(z)=0.$
Am I right?