Order of Poles

Complex Analysis · Axiom Academy

One number that says exactly how violently a function blows up — and which residue formula you'll need. 1. Order Measures the Strength of the Blow-Up An isolated singularity z_0 is a pole when as . The order m records how fast . Watch three poles at the origin as a probe slides inward: climbs steadily, climbs faster, and rockets up. Same singular point, three different strengths . Order 1 — a simple pole, the gentlest blow-up Higher order — a steeper, more violent blow-up 2. The Limit Test: Tame It with Factors of (z-z_0) To find the order, multiply f by (z-z_0)^ m and watch the limit. Too few factors and it's still infinite; the right number tames the blow-up to a finite, nonzero value; one too many and it collapses to 0 . The animation peels factors off f(z)=1/z^ 3 : the spike shrinks step by step until at m=3 the curve flattens to the height 1 . — keep going, the pole is stronger than m . The limit is finite and nonzero . That smallest m is the order of the pole. The limit is 0 — you over-multiplied; the order was m-1 . Order = the smallest positive integer m for which the limit exists and is nonzero. At m=1 and m=2 the limit is still . At m=3 , , finite and nonzero — so this is a pole of order 3 . (For a quotient P/Q you can shortcut: order = zero-order of Q minus zero-order of P .) 3. The Laurent Fingerprint Reads the Order Off Directly

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