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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Laurent series canonical | 2 |
How this entity was disambiguated
This entity first appeared as the object of triple T6910707 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Laurent series Context triple: [Hahn series, generalizes, Laurent series]
-
A.
Taylor series
A Taylor series is an infinite sum of terms calculated from the derivatives of a function at a single point, used to represent and approximate functions as power series.
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B.
Lambert series
Lambert series are special infinite series in number theory and analysis, often involving arithmetic functions and powers of a variable, introduced by Johann Heinrich Lambert and used in the study of modular forms and q-series.
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C.
Dirichlet series
A Dirichlet series is an infinite series of the form ∑ aₙ/nˢ, fundamental in analytic number theory for studying arithmetic functions and L-functions.
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D.
Cauchy–Hadamard theorem
The Cauchy–Hadamard theorem is a fundamental result in complex analysis that characterizes the radius of convergence of a power series in terms of the growth rate of its coefficients.
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E.
Mittag-Leffler function
The Mittag-Leffler function is a complex function that generalizes the exponential function and plays a central role in fractional calculus and the theory of differential and integral equations.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Laurent series Target entity description: 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.
-
A.
Taylor series
A Taylor series is an infinite sum of terms calculated from the derivatives of a function at a single point, used to represent and approximate functions as power series.
-
B.
Lambert series
Lambert series are special infinite series in number theory and analysis, often involving arithmetic functions and powers of a variable, introduced by Johann Heinrich Lambert and used in the study of modular forms and q-series.
-
C.
Dirichlet series
A Dirichlet series is an infinite series of the form ∑ aₙ/nˢ, fundamental in analytic number theory for studying arithmetic functions and L-functions.
-
D.
Cauchy–Hadamard theorem
The Cauchy–Hadamard theorem is a fundamental result in complex analysis that characterizes the radius of convergence of a power series in terms of the growth rate of its coefficients.
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E.
Mittag-Leffler function
The Mittag-Leffler function is a complex function that generalizes the exponential function and plays a central role in fractional calculus and the theory of differential and integral equations.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Laurent series Description of subject: 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.
Referenced by (2)
Full triples — surface form annotated when it differs from this entity's canonical label.