Classifying Discontinuity of 1/z

Complex Analysis · Axiom Academy

EXAMPLE Classifying the Discontinuity of 1/z Working out why the singularity of f(z) = 1/z at z = 0 is a simple pole The function is undefined at z = 0 . Classify the singularity there: is it removable , a pole (and of what order), or essential ? Nice work. You classified the singularity of at z = 0 from the ground up — checking each possibility in turn. Isolated & not removable: f is analytic on a punctured disk about 0 , but is not finite, so the singularity can't be removed. It's a pole: as — the hallmark of a pole, not an essential singularity. Order 1 (simple pole): is finite and nonzero at 0 ; equivalently the Laurent series is just , a single negative-power term. A simple pole is the mildest kind of pole. The residue here is 1 — the value you'll feed into contour integration later. The map sends a small circle of radius r around the origin to a large circle of radius : as the modulus . The blow-up is controlled and algebraic — the signature of a pole, not the wild every-direction behavior of an essential singularity.

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