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Definition: 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.