Residues at Higher Order Poles

Complex Analysis · Axiom Academy

LESSON Residues at Higher Order Poles One coefficient hides inside a Laurent series — here is the machine that digs it out at a pole of any order. 1. The Residue Is One Hidden Coefficient Near a pole of order m at z_0 , the function splits into a principal part (the blow-up terms with negative powers) plus an ordinary power series: The residue is exactly the coefficient of A spotlight scans the Laurent terms and locks onto the one that is the residue. Multiply f by (z-z_0)^m . Every term's power jumps up by m , so all the negative powers become non-negative — the pole is gone and what remains is an ordinary analytic function g(z) : Each term lifts by m powers; the residue term lands exactly at power m-1 . a_ -m /(z-z_0)^m becomes the constant a_ -m — no more blow-up. g is a clean Taylor series. a_ -1 /(z-z_0) becomes a_ -1 (z-z_0)^ m-1 — the residue now sits at power m-1 of g . 3. Differentiate, Then Read It Off The coefficient of (z-z_0)^ m-1 in a power series is exactly what the (m-1) th derivative — evaluated at z_0 and divided by (m-1)! — pulls out. Lower terms vanish under the derivative; higher terms vanish in the limit. Only a_ -1 survives: Differentiating m-1 times erases the lower terms; the limit kills the higher ones — leaving a_ -1 . Worked check — a double pole ( m = 2 ) Take , a pole of order 2 at z_0=0 . Clear it: . Differentiate once and take the limit: The factor here is , so . For a triple pole ( m=3 ) you would differentiate twice and divide by 2! — e.g. .

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