A simple pole of an analytic function f is a pole of order one. That is, (z - z_0) f(z) is an analytic function at the pole z = z_0. Alternatively, its principal part is c/(z - z_0) for some c!=0. It is called simple because a function with a pole of order n at a can be written as the product of n functions with simple poles at z_0.

essential singularity | pole