Isolated singularity

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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 |f| tends to infinity as f tends to a; in this case some power of z-a times f is bounded, and the singularity is a pole.
  • Neither of the above occurs, and the singularity is essential.