For the following, determine if the given point is a removable singularity, an essential singularity, or a pole (indicate its order).

a. \(\frac{1-\cos z}{z^{2}}, \quad z=0\).

b. \(\frac{\sin z}{z^{2}}, \quad z=0\).

c. \(\frac{z^{2}-1}{(z-1)^{2}}, \quad z=1\)

d. \(z e^{1 / z}, \quad z=0\).

e. \(\cos \frac{\pi}{z-\pi}, \quad z=\pi\).