# Isolated singularity

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Revision as of 18:26, 28 August 2011 by Boris Tsirelson (talk | contributions)

In complex analysis, an **isolated singularity** of a complex-valued function is a point at which the function is not holomorphic, but which has a neighbourhood on which the function is holomorphic.

Suppose that *f* is holomorphic on a neighbourhood *N* of *a* except possibly at *a*. The behaviour of the function can be of one of three types:

- The absolute value of
*f*is bounded on*N*; in this case*f*tends to a limit at*a*, and the singularity is removable. - The absolute value of
*f*tends to infinity as*f*tends to*a*; in this case some power of*z*-*a*times*f*(*z*) is bounded, and the singularity is a pole. - Neither of the above occurs, and the singularity is essential.

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