Finding the Order of a Pole

Complex Analysis · Axiom Academy

EXAMPLE Finding the Order of a Pole Classify every pole of a rational function — and don't get fooled by a numerator that cancels. Find every pole of and state its order. The denominator hands us the candidate locations, but the numerator gets the final say. We'll factor, hunt for cancellation, read off the orders, and confirm with the limit test. Nice work. You classified both singularities of a rational function and dodged the cancellation trap that makes the answer different from "whatever power is in the denominator." Factor first, then check the numerator: a singularity is only a pole of order m if the numerator does not vanish there. Cancellation lowers the order: the numerator's factor (z-2) knocked the apparent order at z=2 down from 3 to 2 . The order formula: at a point, . The limit test confirms it: the order is the smallest m for which is finite and nonzero. Result: f has a double pole at z=2 and a simple pole at z=-1 . Knowing each pole's order is exactly what you need next: the residue formula and Cauchy's residue theorem both depend on it.

This is the written version of the interactive lesson above. See the full Complex Analysis course.