Laurent series

E627725

complex analysis concept mathematical concept

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.

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All labels observed (1)

Label	Occurrences
Laurent series canonical	3

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.
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.
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.
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 ⓘ
principalPart	∑_{n=1}^{∞} a_{-n} (z - z₀)^{-n} ⓘ
regularPart	∑_{n=0}^{∞} a_n (z - z₀)^n ⓘ
relatedTo	Cauchy integral formula ⓘ Taylor series expansion ⓘ residue theorem ⓘ linked to: Cauchy residue theorem
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 ⓘ