Laurent series
E627725
A Laurent series is a representation of a complex function as a power series that can include terms with negative as well as nonnegative integer powers of the variable, typically used to describe behavior near singularities.
Statements (47)
| Predicate | Object |
|---|---|
| instanceOf |
complex analysis concept
ⓘ
mathematical concept ⓘ |
| allows | representation of functions with isolated singularities inside the annulus ⓘ |
| appliesTo | functions analytic on an annulus ⓘ |
| canBe | finite in the negative direction for meromorphic functions ⓘ |
| centeredAt | a point z₀ in the complex plane ⓘ |
| characterizes |
essential singularities
ⓘ
poles ⓘ removable singularities ⓘ |
| coefficientComputation | a_n = (1 / 2πi) ∮_C f(z) (z - z₀)^{-n-1} dz ⓘ |
| coefficientInterpretation | a_{-1} is the residue at z₀ ⓘ |
| convergenceRegion | {z : r < |z - z₀| < R} ⓘ |
| convergesOn | an annulus around the center ⓘ |
| criterionForEssentialSingularity | infinitely many negative-power coefficients are nonzero ⓘ |
| criterionForPole | finitely many negative-power coefficients are nonzero ⓘ |
| criterionForRemovableSingularity | all negative-power coefficients are zero ⓘ |
| describes | complex functions ⓘ |
| domain | complex variable z ⓘ |
| example |
1/(z(z-1)) has a Laurent expansion with a pole at z=0
ⓘ
1/z = ∑_{n=0}^{∞} (-1)^n (z-1)^n for |z-1|<1, written as Laurent series around z₀=1 ⓘ |
| expansionPoint | often chosen at an isolated singularity ⓘ |
| field |
complex analysis
ⓘ
mathematical analysis ⓘ |
| generalizes | Taylor series ⓘ |
| hasComponent |
principal part
ⓘ
regular part ⓘ |
| hasConstraint | coefficients are complex numbers ⓘ |
| hasForm | ∑_{n=-∞}^{∞} a_n (z - z₀)^n ⓘ |
| hasHistoricalNote | introduced in the 19th century ⓘ |
| hasProperty | is a power series with possibly negative integer powers ⓘ |
| implies | local representation of analytic functions on annuli ⓘ |
| is | a doubly infinite series in general ⓘ |
| isToolIn |
local analysis of complex functions
ⓘ
meromorphic function theory ⓘ |
| namedAfter | Pierre Alphonse Laurent NERFINISHED ⓘ |
| principalPart | ∑_{n=1}^{∞} a_{-n} (z - z₀)^{-n} ⓘ |
| regularPart | ∑_{n=0}^{∞} a_n (z - z₀)^n ⓘ |
| relatedTo |
Cauchy integral formula
NERFINISHED
ⓘ
Taylor series expansion ⓘ residue theorem NERFINISHED ⓘ |
| uniquenessProperty | expansion is unique in its annulus of convergence ⓘ |
| usedFor |
applying the residue theorem
ⓘ
classifying singularities ⓘ computing residues ⓘ evaluating complex integrals ⓘ representing functions near singularities ⓘ studying isolated singularities ⓘ |
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.