KnowraLaurent seriesLinked fromLinked fromThe 21 pages that link to Laurent series, each with the reason it gives.All 21Broader topic 3Related 11Narrower topic 1Compared with 6Taylor seriesCompared with: Laurent series can represent functions near singularities where an ordinary Taylor series cannot.Complex analysisRelated: Its negative-power terms describe behavior near isolated singularities.Power seriesCompared with: Unlike an ordinary power series, it can represent functions near isolated singularities.Analytic continuationRelated: It represents continuations near isolated singularities where Taylor series fail.Cauchy's integral formulaRelated: The formula’s extension to punctured regions is closely connected to Laurent expansions.Meromorphic functionRelated: Its negative-power terms encode a meromorphic function's behavior near each pole.Taylor's theoremCompared with: It handles isolated singularities where an ordinary Taylor expansion may fail.Formal power seriesRelated: It extends power-series algebra to include finitely many negative powers.Residue theoremRelated: Its coefficient of the −1 power is the local quantity summed by the theorem.ResidueNarrower topic: Its negative-power terms encode the local singular behavior from which a residue is read.Removable singularityRelated: The singularity is removable exactly when all negative-power terms vanish.Taylor polynomialCompared with: It can represent singular behavior that an ordinary Taylor polynomial cannot capture.Mittag-Leffler theoremBroader topic: Laurent expansions encode each prescribed principal part near its pole.Jacobi triple productRelated: The bilateral expansion is naturally a Laurent series in the variable.Binomial seriesCompared with: The binomial series uses only nonnegative powers of \(x\), unlike expansions around punctured neighborhoods.Hartogs extension theoremRelated: In one variable, negative powers describe isolated singularities that obstruct extension.Gösta Mittag-LefflerRelated: Its principal parts describe the local behavior that the Mittag-Leffler theorem assembles globally.Casorati–Weierstrass theoremRelated: An essential singularity is characterized by infinitely many negative-power terms in its Laurent series.Analytic functionBroader topic: Near isolated singularities, analytic behavior is described by Laurent rather than Taylor series.Analyticity of holomorphic functionsCompared with: When a holomorphic Taylor expansion fails at an isolated singularity, Laurent terms describe the local behavior.Series expansionBroader topic: It extends power-series representation to neighborhoods containing isolated singularities.
KnowraLaurent seriesLinked fromLinked fromThe 21 pages that link to Laurent series, each with the reason it gives.All 21Broader topic 3Related 11Narrower topic 1Compared with 6Taylor seriesCompared with: Laurent series can represent functions near singularities where an ordinary Taylor series cannot.Complex analysisRelated: Its negative-power terms describe behavior near isolated singularities.Power seriesCompared with: Unlike an ordinary power series, it can represent functions near isolated singularities.Analytic continuationRelated: It represents continuations near isolated singularities where Taylor series fail.Cauchy's integral formulaRelated: The formula’s extension to punctured regions is closely connected to Laurent expansions.Meromorphic functionRelated: Its negative-power terms encode a meromorphic function's behavior near each pole.Taylor's theoremCompared with: It handles isolated singularities where an ordinary Taylor expansion may fail.Formal power seriesRelated: It extends power-series algebra to include finitely many negative powers.Residue theoremRelated: Its coefficient of the −1 power is the local quantity summed by the theorem.ResidueNarrower topic: Its negative-power terms encode the local singular behavior from which a residue is read.Removable singularityRelated: The singularity is removable exactly when all negative-power terms vanish.Taylor polynomialCompared with: It can represent singular behavior that an ordinary Taylor polynomial cannot capture.Mittag-Leffler theoremBroader topic: Laurent expansions encode each prescribed principal part near its pole.Jacobi triple productRelated: The bilateral expansion is naturally a Laurent series in the variable.Binomial seriesCompared with: The binomial series uses only nonnegative powers of \(x\), unlike expansions around punctured neighborhoods.Hartogs extension theoremRelated: In one variable, negative powers describe isolated singularities that obstruct extension.Gösta Mittag-LefflerRelated: Its principal parts describe the local behavior that the Mittag-Leffler theorem assembles globally.Casorati–Weierstrass theoremRelated: An essential singularity is characterized by infinitely many negative-power terms in its Laurent series.Analytic functionBroader topic: Near isolated singularities, analytic behavior is described by Laurent rather than Taylor series.Analyticity of holomorphic functionsCompared with: When a holomorphic Taylor expansion fails at an isolated singularity, Laurent terms describe the local behavior.Series expansionBroader topic: It extends power-series representation to neighborhoods containing isolated singularities.