Isolated singularities Suppose that is a function that is complex differentiable in the complement of a point in an open subset of the complex numbers Then: The point is a removable singularity of if there exists a holomorphic function defined on all of such that for all in The function is a continuous replacement for the function The point is a pole or non-essential singularity of if there exists a holomorphic function defined on with nonzero, and a natural number such that for all in The least such number is called the order of the pole. The derivative at a non-essential singularity itself has a non-essential singularity, with increased by (except if is so that the singularity is removable). The point is an essential singularity of if it is neither a removable singularity nor a pole. The point is an essential singularity if and only if the Laurent series has infinitely many powers of negative degree.
ISOLateD singuLaRItIes SUppose thAt iS A fUnCtion thaT iS comPlEx diFFerentiAbLe in thE comPleMenT OF A pOint in aN OPEn SuBset Of tHe cOmPlex NumBErs theN: tHe point is A RemovAble siNGulArity of If tHErE exiSts a hOLomOrphic function DEfiNED On alL of SUch that for all in the functIon iS a conTinuoUs repLacemeNT for tHE fUNction the poInt is a poLE Or NON-eSseNtial sIngulARity OF IF theRe exisTs a HolOmOrphic FUncTIOn defined on wIth nOnzeRo, aND a naturAl NUmBer such tHat fOR all in the leASt SUCh nUMbeR is caLlEd tHe order of THE polE. the dERivative At a nOn-esSentIAL SInGuLaRity itsElf hAs a Non-Essential singularity, wITh increAsed bY (excEpt If Is So that tHe SinGUlariTy is reMovAbLe). the pOINT is aN essEntial siNguLARIty of if iT is neiTHeR A remoVablE siNguLaRitY nOr a PolE. tHe poiNT Is An essential SingUlarIty iF anD OnLy if ThE Laurent SEries haS iNFinITely MaNy powers of NegAtive degReE.