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Power series with negative powers
mathematics, the Laurent series of a complex function f ( z ) {\displaystyle f(z)} is a representation of that function as a power series which includes
Laurent_series
Infinite sum that is considered independently from any notion of convergence
power series Puiseux series are an extension of formal Laurent series, allowing fractional exponents Rational series Ring of restricted power series International
Formal_power_series
Widely-used term in mathematics
independent meanings but usually refers to the negative-power portion of the Laurent series of a function. The principal part at z = a {\displaystyle z=a} of a
Principal_part
Mathematical approximation of a function
analytic. Some singularities, namely poles, can be accounted for by a Laurent series. If f {\displaystyle f} has a pole of order k {\displaystyle k} at z
Taylor_series
Name list
Hermann Laurent, French mathematician Pierre Alphonse Laurent, French mathematician best known as the discoverer of the Laurent series Auguste Laurent (1807–1853)
Laurent_(name)
Attribute of a mathematical function
by finding Laurent series expansions, and one can define the residue as the coefficient a − 1 {\displaystyle a_{-1}} of a Laurent series. The concept
Residue_(complex_analysis)
Expression of a function as an infinite sum of simpler functions
The Maclaurin series of a function f is a special case of its Taylor series about x 0 = 0 {\displaystyle x_{0}=0} . A Laurent series is a generalization
Series_expansion
Extremely small quantity in calculus; thing so small that there is no way to measure it
of Dales and Woodin. Since a Taylor series evaluated with a Laurent series as its argument is still a Laurent series, the system can be used to do calculus
Infinitesimal
French mathematician (1813–1854)
Alphonse Laurent (18 July 1813 – 2 September 1854) was a French mathematician, engineer, and Military Officer best known for discovering the Laurent series, an
Pierre_Alphonse_Laurent
Power series with rational exponents
Puiseux series becomes a Laurent series in an nth root of the indeterminate. For example, the example above is a Laurent series in x 1 / 6 . {\displaystyle
Puiseux_series
Topics referred to by the same term
minor planet (51) Nemausa Laurent, South Dakota, a proposed town for the Deaf to be named for Laurent Clerc Laurent series, in mathematics, representation
Laurent
Analytic function in mathematics
of order one at s = 1. It can therefore be expanded as a Laurent series about s = 1; the series development is then ζ ( s ) = 1 s − 1 + ∑ n = 0 ∞ γ n n
Riemann_zeta_function
Special function in mathematics
powers of integers. The Laurent series expansion can be used to define generalized Stieltjes constants that occur in the series ζ ( s , a ) = 1 s − 1 +
Hurwitz_zeta_function
Infinite sum
formal power series is also a differential algebra, with differentiation performed term-by-term. Laurent series generalize power series by admitting terms
Series_(mathematics)
Concept of complex analysis
{\displaystyle a_{-1}} of ( z − c ) − 1 {\displaystyle (z-c)^{-1}} in the Laurent series expansion of f {\displaystyle f} around c {\displaystyle c}
Residue_theorem
Matrix operation generalizing exponentiation of scalar numbers
Qa,t(z) is solved from the product of P by the principal part of the Laurent series of f at a: It is proportional to the relevant Frobenius covariant. Then
Matrix_exponential
French mathematician
of works, Laurent series expansions for complex functions were not named after him, but after Pierre Alphonse Laurent. Théorie des séries, contenant
Paul_Matthieu_Hermann_Laurent
Infinite sum of monomials
x^{-1}+1+x^{1}+x^{2}+\cdots } is not considered a power series (although it is a Laurent series). Similarly, fractional powers such as x 1 2 {\textstyle
Power_series
Way of writing a meromorphic function
the polynomial terms contributed are exactly the regular part of the Laurent series up to z p {\displaystyle z^{p}} . For the other poles λ k {\displaystyle
Partial fractions in complex analysis
Partial_fractions_in_complex_analysis
Algebraic structure also called skew field
conjugation), then the resulting ring of Laurent series is a noncommutative division ring known as a skew Laurent series ring; if σ = id then it features the
Division_ring
Algebraic structure with addition, multiplication, and division
power series (in which k ≥ 0). Since any Laurent series is a fraction of a power series divided by a power of x (as opposed to an arbitrary power series),
Field_(mathematics)
Branch of mathematics studying functions of a complex variable
essential singularities are called meromorphic. Laurent series are the complex-valued equivalent to Taylor series, but can be used to study the behavior of
Complex_analysis
Rational number sequence
&0<&|x|<\pi .\end{aligned}}} The Bernoulli numbers appear in the following Laurent series: Digamma function: ψ ( z ) = ln z − ∑ k = 1 ∞ B k + k z k {\displaystyle
Bernoulli_number
Decomposition of periodic functions
cosine series Fourier transform Gibbs phenomenon Half-range Fourier series Laurent series – the substitution q = eix transforms a Fourier series into a
Fourier_series
Has no other singularities close to it
function is meromorphic. Many important tools of complex analysis such as Laurent series and the residue theorem require that all relevant singularities of the
Isolated_singularity
Polynomial with only one term
the context of Laurent polynomials and Laurent series, the exponents of a monomial may be negative, and in the context of Puiseux series, the exponents
Monomial
Methods of estimating the doubling time of an investment
(ln) from BetterExplained "Taylor series of 1 / (1 – r/200)". WolframAlpha. Retrieved January 3, 2025. "Laurent series of ln(2) / ln(1 + r/100)". WolframAlpha
Rule_of_72
Locally compact infinite topological field
field F q ( ( t ) ) {\displaystyle \mathbb {F} _{q}((t))} of formal Laurent series in the variable t {\displaystyle t} over a finite field F q {\displaystyle
Local_field
Polynomial with negative exponents
In mathematics, a Laurent polynomial (named after Pierre Alphonse Laurent) in one variable over a field F {\displaystyle \mathbb {F} } is a linear combination
Laurent_polynomial
French actress
Claire Laurent in the Netflix series Lupin. Makowski had her breakthrough acting role in 2021 when she was cast as the teenaged version of Claire Laurent (played
Ludmilla_Makowski
p-adic numbers Qp (where p is any prime number), or the field of formal Laurent series Fq((T)) over a finite field Fq. Local class field theory gives a description
Local_class_field_theory
Linear transform from the time domain to the frequency domain
Z-transform can also be viewed as a Laurent series where one views the sequence of numbers under consideration as the (Laurent) expansion of an analytic function
Z-transform
American film director
Laurent Touil-Tartour (born November 23, 1971) is a French film director, screenwriter, and producer. He is known for writing, directing and producing
Laurent_Touil-Tartour
Type of mathematical functions
studied are holomorphic or complex analytic so that, locally, they are power series in the variables zi. Equivalently, they are locally uniform limits of polynomials;
Function of several complex variables
Function_of_several_complex_variables
Location around which a function displays irregular behavior
. Another way to characterize an essential singularity is that the Laurent series of f {\displaystyle f} at the point a {\displaystyle a} has infinitely
Essential_singularity
French actress, director and singer (born 1983)
Mélanie Laurent (French pronunciation: [melani loʁɑ̃] ; born 21 February 1983) is a French actress, filmmaker, and singer. She has received two César
Mélanie_Laurent
2015 British historical drama television series
Ellenborough (series 5) Dan O'Keefe as Coldbath Prison Guard (series 5) Don Gallagher as Vicar (series 5) Zachary Fall as Laurent (series 5) Nico Rogner
Poldark_(2015_TV_series)
Type of function in mathematics
\mathbb {Q} _{p}} and its finite extension fields, and fields of formal Laurent series F q ( ( t ) ) {\displaystyle \mathbb {F} _{q}((t))} over a finite field
Analytic_function
French actor and director (born 1973)
Laurent Fabien Fernand Lafitte (French pronunciation: [lɔʁɑ̃ lafit]; born 22 August 1973) is a French actor. He was nominated three times for the César
Laurent_Lafitte
French racing driver (born 1998)
Christophe Pierre-François Laurent (born 5 April 1998) is a French racing driver who currently competes in the 2025 European Le Mans Series with DKR Engineering
Thomas_Laurent
Tool for solving polynomial equations
essentially the field of formal Laurent series in the indeterminate X, i.e. the field of fractions of the formal power series ring K [ [ X ] ] {\displaystyle
Newton_polygon
Concept in complex analysis
{\displaystyle z_{0},} a nonzero meromorphic function f is the sum of a Laurent series with at most finite principal part (the terms with negative index values):
Zeros_and_poles
Difference between logarithm and harmonic series
x-\gamma } . Evaluations of the digamma function at rational values. The Laurent series expansion for the Riemann zeta function*, where it is the first of the
Euler's_constant
Point where a mathematical object behaves irregularly
a {\displaystyle a} is an essential singularity if and only if the Laurent series has infinitely many powers of negative degree. Other than isolated singularities
Singularity_(mathematics)
Prime Minister of Canada from 1948 to 1957
Louis Stephen St. Laurent (French: [lwi sɛ̃ lɔʁɑ̃]; February 1, 1882 – July 25, 1973) was a Canadian lawyer and politician who served as the 12th prime
Louis_St._Laurent
Jamaican-British actor
performance in Red Pitch. In June 2023, Lovehall played Laurent in musical television series Champion by Candice Carty-Williams, for BBC One. That year
Francis_Lovehall
Mathematical formal infinite series
complete metric space. However, unlike in the case of formal Laurent series or Puiseux series, the formal sums used in defining the elements of the field
Hahn_series
Multivalued function in mathematics
L2 = ln(−ln(−x)). Integer powers of W0 also admit simple Taylor (or Laurent) series expansions at zero: W 0 ( x ) 2 = ∑ n = 2 ∞ − 2 ( − n ) n − 3 ( n −
Lambert_W_function
Summability method in physics
zeta-regularizations are related. By expanding the exponential sum as a Laurent series F ( t ) = a N t N + a N − 1 t N − 1 + ⋯ {\displaystyle F(t)={\frac
Zeta_function_regularization
Topics referred to by the same term
"approximately open" and "approximately closed" The regular part, of a Laurent series, the series of terms with positive powers Regular singular points, in theory
Regular
Method for solving ordinary differential equations
approach to actually calculate the series coefficients in all cases. Fuchs' theorem Regular singular point Laurent series Weisstein, Eric W. "Frobenius Method"
Frobenius_method
In mathematics, the Faber polynomials Pm of a Laurent series f ( z ) = z − 1 + a 0 + a 1 z + ⋯ {\displaystyle \displaystyle f(z)=z^{-1}+a_{0}+a_{1}z+\cdots
Faber_polynomials
Mathematical series
include ordinary power series, Laurent series, Fourier series, Liouville-Neumann series, formal power series, and Puiseux series. There exist many types
Function_series
Maclaurin series of sine, cosine and arctangent
power series expansion for tan−1 q above. Madhava of Sangamagrama Madhava's sine table Madhava's correction term Padé approximant Taylor series Laurent series
Madhava_series
Second-order partial differential equation
the solution of known regions in Laurent series (about r = ∞ {\displaystyle r=\infty } ), instead of Taylor series (about r = 0 {\displaystyle r=0} )
Laplace's_equation
Infinite series that is not convergent
singularity, the sum is defined by the constant term of the Laurent series expansion. If the series f ( s ) = 1 a 1 s + 1 a 2 s + 1 a 3 s + ⋯ {\displaystyle
Divergent_series
Canadian ice hockey player (born 1993)
Laurent Brossoit (born March 23, 1993) is a Canadian professional ice hockey player who is a goaltender for the San Diego Gulls in the American Hockey
Laurent_Brossoit
Mélanie Laurent is a French actress, singer, screenwriter and director. She initially rose to prominence for her performance in the 2006 French drama
List of roles and awards of Mélanie Laurent
List_of_roles_and_awards_of_Mélanie_Laurent
(of a Laurent series) consists of the series of terms with positive powers
In mathematics, the regular part of a Laurent series consists of the series of terms with positive powers. That is, if f ( z ) = ∑ n = − ∞ ∞ a n ( z −
Regular_part
recurrence relations. The field involves bijections, power series and formal Laurent series. Gessel, Ira M.; Stanley, Richard P. (1995), "Algebraic enumeration"
Algebraic_enumeration
Constants in the zeta function's Laurent series expansion
constants are the numbers γ k {\displaystyle \gamma _{k}} that occur in the Laurent series expansion of the Riemann zeta function: ζ ( 1 + s ) = 1 s + ∑ n = 0
Stieltjes_constants
Infinite sum of positive and negative powers of a formal variable
appropriate space of test functions. They are closely related to formal Laurent series, but are not required to have finitely many negative powers. In particular
Formal_distribution
President of the DRC from 1997 to 2001
Laurent-Désiré Kabila (French pronunciation: [lo.ʁɑ̃ de.zi.ʁe ka.bi.la]; 27 November 1939 – 16 January 2001), usually known as Laurent Kabila or Kabila
Laurent-Désiré_Kabila
Complex-differentiable (mathematical) function
function is infinitely differentiable and locally equal to its own Taylor series (is analytic). Holomorphic functions are the central objects of study in
Holomorphic_function
Hyperbolic analogues of trigonometric functions
integration. It is possible to express explicitly the Taylor series at zero (or the Laurent series, if the function is not defined at zero) of the above functions
Hyperbolic_functions
Semiring with minimum and addition replacing addition and multiplication
{Q} } , the field of formal Laurent series K ( ( t ) ) {\displaystyle K((t))} (integer powers), or the field of Puiseux series K { { t } } {\displaystyle
Tropical_semiring
Number system extending the rational numbers
directly from the fundamental theorem of arithmetic. A p-adic series is a formal Laurent series of the form ∑ i = v ∞ r i p i , {\displaystyle \sum _{i=v}^{\infty
P-adic_number
Concept in abstract algebra
{\mathbb {Z} }}} and so has cohomological dimension 1. The field of formal Laurent series k ( ( t ) ) {\displaystyle k((t))} over an algebraically closed field
Cohomological_dimension
Skeletonized version of algebraic geometry
The field of Laurent series C ( ( t ) ) {\displaystyle \mathbb {C} (\!(t)\!)} (integer powers), or the field of (complex) Puiseux series C { { t } } {\displaystyle
Tropical_geometry
Concept in algebraic number theory
is an integer. The q-expansion of j, with its Fourier series expansion written as a Laurent series in terms of q = e 2 π i τ {\displaystyle q=e^{2\pi i\tau
Heegner_number
coefficients q 1 , … , q n {\displaystyle q_{1},\dots ,q_{n}} be expandable as Laurent series with finite principle part at ξ {\displaystyle \xi } . Then there exists
Fuchsian_theory
Method of evaluating certain integrals along paths in the complex plane
{-5}{64}}(z-i)^{2}+\cdots } (See the sample Laurent calculation from Laurent series for the derivation of this series.) It is clear by inspection that the residue
Contour_integration
Algorithm to approximate functions
lemma – TheoremPages displaying short descriptions with no spaces Laurent series – Power series with negative powers Padé approximant – 'Best' approximation
Remez_algorithm
French pornographic actress (born 1973)
Draghixa Laurent (born 3 June 1973), best known as Draghixa, is a French former pornographic actress and singer. Born in Split, Croatia, Draghixa moved
Draghixa_Laurent
Mathematical theorem
norm, so a finite extension of the p-adic numbers Qp or of the formal Laurent series Fq((T)); they also handle the adelic case in characteristic 0, combining
Selberg_trace_formula
Infinite dimensional Lie group
rules given by an algebra homomorphism Φ of H into the algebra V of Laurent series in z with poles of finite order; a renormalization scheme given by a
Butcher_group
Mathematical function
_{0}^{1}{\frac {x^{z-1}\ln {x}}{1-x}}\,dx} An asymptotic expansion as a Laurent series can be obtained via the derivative of the asymptotic expansion of the
Trigamma_function
Theorem in complex analysis
ISBN 978-0-521-80937-5 Ahlfors, Lars (2000), Complex Analysis, McGraw-Hill series in Mathematics, McGraw-Hill, ISBN 0-07-000657-1 Lang, Serge (2003), Complex
Cauchy's_integral_theorem
2018 French television series
Dujardin as Thierry Personnaz (child) Eliott Lobrot as Laurent Personnaz (child) Charles Petit as Laurent Personnaz (adult) Catherine Vinatier as Doctor Ségur
The_Chalet_(TV_series)
Computer algebra system
and limits. It can derive closed-form series expansions as well as terms of Taylor-Maclaurin-Laurent series. It can perform matrix manipulations with
Maxima_(software)
Lists of values of mathematical functions
approximation, and typically for higher or variable precisions, Taylor and Laurent series) with range reduction and a table lookup — they first look up the closest
Trigonometric_table
Theorem in complex analysis
{\displaystyle f} is an entire function, it can be represented by its Taylor series about 0: f ( z ) = ∑ k = 0 ∞ a k z k {\displaystyle f(z)=\sum _{k=0}^{\infty
Liouville's theorem (complex analysis)
Liouville's_theorem_(complex_analysis)
Theorem in complex analysis
centered at a {\displaystyle a} it can be expanded as a convergent power series f ( z ) = ∑ n = 0 ∞ c n ( z − a ) n {\displaystyle f(z)=\sum _{n=0}^{\infty
Analyticity of holomorphic functions
Analyticity_of_holomorphic_functions
Extension of the domain of an analytic function (mathematics)
further values of a function, for example in a new region where the infinite series representation that initially defined the function becomes divergent. The
Analytic_continuation
case of complex quadratic polynomial one can compute this map using Laurent series about infinity c = Ψ M ( w ) = w + ∑ m = 0 ∞ b m w − m = w − 1 2 + 1
External_ray
Home video game consoles
The Xbox Series X and Xbox Series S are the fourth generation of consoles in Microsoft's Xbox series, succeeding the previous generation's Xbox One. Released
Xbox_Series_X_and_Series_S
Numerical analysis of electric power flow
LPF is based on the current injection method (CIM) and applies the Laurent series expansion. The main characteristics of this formulation are its proven
Power-flow_study
Mathematical transformation
{1}{z^{n+1}}}\right).} Under certain conditions the complete expansion as a Laurent series can be obtained: S ρ ( z ) = ∑ n = 0 ∞ m n z n + 1 . {\displaystyle
Stieltjes_transformation
Approximation of a function by a polynomial
lemma – TheoremPages displaying short descriptions with no spaces Laurent series – Power series with negative powers Padé approximant – 'Best' approximation
Taylor's_theorem
Analytic function on the upper half-plane with a certain behavior under the modular group
function can only have a finite number of negative-exponent terms in its Laurent series, its q-expansion. It can only have at most a pole at q = 0, not an essential
Modular_form
in the product of two formal Laurent series. Thus, given a polynomial p(t) with leading term tN, the formal Laurent series τ j − N p ( τ − 1 ) = τ j τ
Cofree_coalgebra
Concept in abstract algebra
indeterminate x in a field K(x) of rational functions or K((x)) of formal Laurent series is defined to be transcendental over K. Algebraic functions are the
Algebraic_element
Mathematical identity found by Jacobi in 1829
f_{x}} is meromorphic for | y | > 0 {\displaystyle |y|>0} , it has a Laurent series f x ( y ) = ∑ n = − ∞ ∞ c n ( x ) y 2 n {\displaystyle f_{x}(y)=\sum
Jacobi_triple_product
French police television series
television series created by Lionel Olenga, Laurent Scalese and Stéphane Drouet, broadcast since October 25, 2013 on France 2. In 2019, the series was canceled
Cherif_(TV_series)
1867 novel by Émile Zola
turbulent, sordid affair with Camille's friend, Laurent. Despite their numerous trysts, Thérèse and Laurent are convinced they can only be truly happy if
Thérèse_Raquin
Fluid dynamics phenomenon
|z|=1} , and by representing the functions f , g {\displaystyle f,g} as Laurent series: f ( z ) = ∑ n = − ∞ ∞ f n z n , g ( z ) = ∑ n = − ∞ ∞ g n z n , {\displaystyle
Stokes'_paradox
Square matrix in which each ascending skew-diagonal from left to right is constant
matrix, with symmetric and upper anti-triangular blocks. Given a formal Laurent series f ( z ) = ∑ n = − ∞ N a n z n , {\displaystyle f(z)=\sum _{n=-\infty
Hankel_matrix
Japanese original net animation (ONA) series
various points in the series. He has a thick, peculiar accent when speaking English, occasionally remarked upon by others. Laurent Thierry (ローラン・ティエリー,
Great_Pretender_(TV_series)
Mathematical concept
the affine Grassmannian is a quotient of a group-ring based on formal Laurent series. Given a finite-dimensional vector space V and a non-negative integer
Affine Grassmannian (manifold)
Affine_Grassmannian_(manifold)
Finnish mathematician (1895–1980)
functions analytic in the plane except for isolated points in which the Laurent series of the function has a finite number of terms with a negative power of
Rolf_Nevanlinna
travel, tourism, insurance
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