Residue at Double Pole

Complex Analysis · Axiom Academy

EXAMPLE Residue at a Double Pole Working a residue at a pole of order 2 with the limit-and-derivative formula Find the residue of at the point z_0 = -1 . Nice work. You computed a residue at a double pole end-to-end — multiply away the pole, differentiate once, then take the limit. The recipe (order m ): . For a double pole, m=2 : one derivative, divide by 1!=1 . Don't skip the derivative. For a simple pole you just evaluate; at a double pole the surviving 1/(z-z_0) term means you must differentiate first — forgetting it is the classic mistake. Check the order before you trust it. A factor like (z-z_0)^2 downstairs looks like order 2, but if the numerator also vanishes at z_0 the true order drops. E.g. is a simple pole (residue 1 ), and is order 2 yet has residue 0 . Residues are the payload of the residue theorem: once you can extract this single coefficient at each pole, contour integrals collapse to a sum of these numbers.

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