A Complex function has a pole of order at if, in the Laurent Series, for and . Equivalently, has a pole of order at if is the smallest Positive Integer for which is differentiable at . If , there is no pole at . Otherwise, the order of the pole is the greatest Positive Coefficient in the Laurent Series.
This is equivalent to finding the smallest such that
See also Laurent Series, Residue (Complex Analysis)
Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, pp. 396-397, 1985.