The classification of isolated singularities can be stated through the Laurent expansion of about :
  • The singularity is removable when extends holomorphically to , equivalently when every negative-power Laurent coefficient is zero.
  • It is a pole of order when extends holomorphically to a function nonzero at . Equivalently, the Laurent series starts with a nonzero term and has no more negative power.
  • It is essential when it is neither removable nor a pole, equivalently when infinitely many negative-power coefficients are nonzero.

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