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v. t. e. In mathematics, the Laurent series of a complex function is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied.
Definition. A Laurent polynomial with coefficients in a field is an expression of the form. where is a formal variable, the summation index is an integer (not necessarily positive) and only finitely many coefficients are non-zero. Two Laurent polynomials are equal if their coefficients are equal.
Complex analysis. In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula.
Laurent series definition. The principal part at of a function. is the portion of the Laurent series consisting of terms with negative degree. [ 1] That is, is the principal part of at . If the Laurent series has an inner radius of convergence of , then has an essential singularity at if and only if the principal part is an infinite sum.
A Laurent series is a generalization of the Taylor series, allowing terms with negative exponents; it takes the form = and converges in an annulus. [6] In particular, a Laurent series can be used to examine the behavior of a complex function near a singularity by considering the series expansion on an annulus centered at the singularity.
Complex analysis. In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities. (More generally, residues can be calculated for any function that is holomorphic except at the discrete points { ak } k, even if ...
Stieltjes constants. The area of the blue region converges on the Euler–Mascheroni constant, which is the 0th Stieltjes constant. In mathematics, the Stieltjes constants are the numbers that occur in the Laurent series expansion of the Riemann zeta function : The constant is known as the Euler–Mascheroni constant .
Pierre Alphonse Laurent. Pierre Alphonse Laurent (18 July 1813 – 2 September 1854) was a French mathematician, engineer, and Military Officer best known for discovering the Laurent series, an expansion of a function into an infinite power series, generalizing the Taylor series expansion. He was born in Paris, France.