Isolated Singularities

Complex Analysis · Axiom Academy

A point where a function breaks down — but is surrounded on all sides by good behavior — earns its own private power series. 1. A Bad Point, Ringed by Good Ones A point z_0 is an isolated singularity of f when f fails to be analytic at z_0 , yet is analytic everywhere in a small ring around it — a punctured disk . The singularity stands alone: no other bad points crowd up against it. Analytic on the whole punctured disk — every point except the centre Contrast the two functions on the right. For the only bad point is z_0=0 , and a clean disk of analyticity surrounds it — isolated . For the bad points sit at for every nonzero integer n ; as they pile up on 0 , so no punctured disk around 0 is singularity-free — 0 is not isolated. 2. Isolation Earns a Laurent Series On that punctured disk f can be expanded in powers of (z-z_0) running both ways — a Laurent series . The terms split into a familiar analytic tail (powers ) and a principal part built from the negative powers. The split is the whole point. The non-negative powers form an ordinary analytic function — harmless at z_0 . Everything that makes z_0 singular is packed into the principal part . The coefficient that matters most The single term is special: a_ -1 is the residue of f at z_0 , the one number that survives a loop integral around the point. 3. The Principal Part Names the Type

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