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...
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power series Puiseux series are an extension of formal Laurent series, allowing fractional exponents Rational series Ring of restricted power series The...
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Principal part (section Laurent series definition)
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...
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be calculated by finding Laurent series expansions, and one can define the residue as the coefficient a−1 of a Laurent series. The concept can be used...
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formal power series is also a differential algebra, with differentiation performed term-by-term. Laurent series generalize power series by admitting terms...
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Look up Laurent or laurent in Wiktionary, the free dictionary. Laurent may refer to: Laurent (name), a French masculine given name and a surname Saint...
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Maclaurin series of f is its Taylor series about x 0 = 0 {\displaystyle x_{0}=0} . A Laurent series is a generalization of the Taylor series, allowing...
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Infinitesimal (section Laurent series)
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...
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variable x; see Laurent series. For example, f (x) = e−1/x2 can be written as a Laurent series. The generalization of the Taylor series does converge to...
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Riemann zeta function (redirect from Series of reciprocal powers)
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...
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Alphonse Laurent (18 July 1813 – 2 September 1854) was a French mathematician, engineer, and Military Officer best known for discovering the Laurent series, an...
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Laurent series Auguste Laurent (1807–1853), French chemist Christian Laurent, French engineer Joseph Jean Pierre Laurent, a French astronomer Laurent...
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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...
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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...
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Hurwitz zeta function (section Laurent series)
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 +...
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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...
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Residue theorem (section Series methods)
residue Res(f, c) of f at c is the coefficient a−1 of (z − c)−1 in the Laurent series expansion of f around c. Various methods exist for calculating this...
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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),...
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(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...
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Bernoulli number (section Laurent series)
&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...
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his performance in Red Pitch. In June 2023, Lovehall plays Laurent in musical television series Champion by Candice Carty-Williams, for BBC One. In March...
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In mathematics, a Laurent polynomial (named after Pierre Alphonse Laurent) in one variable over a field F {\displaystyle \mathbb {F} } is a linear combination...
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function is meromorphic. Many important tools of complex analysis such as Laurent series and the residue theorem require that all relevant singularities of the...
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cosine series Fourier transform Gibbs phenomenon Half range Fourier series Laurent series – the substitution q = eix transforms a Fourier series into a...
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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...
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{\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):...
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normalized valuation. The formal Laurent series over a finite field: the ring of integers of Fq((T)) is the ring of formal power series Fq[[T]]. Its maximal ideal...
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Euler's constant (section Series expansions)
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...
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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...
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the solution of known regions in Laurent series (about r = ∞ {\displaystyle r=\infty } ), instead of Taylor series (about r = 0 {\displaystyle r=0} )...
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