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Posted by Herb S. on August 30, 2001 02:42:34 UTC

Singularities: Singularities (in complex function theory) refer to points in the complex plane where the function is not analytic (analytic at a point means it can be represented by a power series convergent in a neighborhood of the point). A pole is a singularity, where the function can be represented by a power series plus a finite series of inverse powers at that point. A simple pole (from what I remember) means that there is only one inverse term of the form c/(z-a), where a is the location of the pole.

The convergence question refers to the series representation of the zeta function. There is a process called analytic continuation, which allows the definition to be extended to all points in the z plane, which are not singularities.

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