Casorati-Weierstrass Theorem

Complex Analysis · Axiom Academy

LESSON The Casorati-Weierstrass Theorem Near an essential singularity a function goes wild — its image comes arbitrarily close to every complex number. 1. Close to Everything You Name Let z_0 be an essential singularity of f . The theorem makes a bold promise: name any complex number w as a target, and pick any tolerance . No matter how tiny a punctured disk you draw around z_0 , there is a point z inside it whose image f(z) lands within of w . for every target w , every tolerance , and every disk radius 2. The Image Stays Dense as the Disk Shrinks Write the punctured disk as . Casorati-Weierstrass says the image is dense in for every . Dense means: every point of is either hit or has image points crowding right up against it. The startling part is what does not happen as . Fix a target w and a small -ball around it. Shrink the disk toward z_0 as far as you like — the coverage never thins out, so that -ball keeps catching an image point. This is precisely how an essential singularity differs from its two tamer cousins: f stays bounded near z_0 — the image lives in one finite blob, far from being dense. , so the image eventually avoids any bounded set — it cannot be dense. The image is dense in — it crowds up against every value at once. 3. Watching e^ 1/z Hit a Target

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