Laurent series

Laurent series (Redirected from Laurent expansion theorem) Jump to navigation Jump to search This article is about doubly infinite power series. For power series with finitely many negative exponents, see Formal Laurent series. A Laurent series is defined with respect to a particular point c and a path of integration γ. The path of integration must lie in an annulus, indicated here by the red color, inside which f(z) is holomorphic (analytic). Mathematical analysis → Complex analysis Complex analysis Complex numbers Real numberImaginary numberComplex planeComplex conjugateUnit complex number Complex functions Complex-valued functionAnalytic functionHolomorphic functionCauchy–Riemann equationsFormal power series Basic Theory Zeros and polesCauchy's integral theoremLocal primitiveCauchy's integral formulaWinding numberLaurent seriesIsolated singularityResidue theoremConformal mapSchwarz lemmaHarmonic functionLaplace's equation Geometric function theory People Augustin-Louis CauchyLeonhard EulerCarl Friedrich GaussJacques HadamardKiyoshi OkaBernhard RiemannKarl Weierstrass Mathematics portal vte In mathematics, the Laurent series of a complex function f(z) 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. The Laurent series was named after and first published by Pierre Alphonse Laurent in 1843. Karl Weierstrass may have discovered it first in a paper written in 1841, but it was not published until after his death.[1] The Laurent series for a complex function f(z) about a point c is given by {estilo de exibição f(z)=soma _{n=-infty }^{infty }uma_{n}(z-c)^{n},} where an and c are constants, with an defined by a line integral that generalizes Cauchy's integral formula: {estilo de exibição a_{n}={fratura {1}{2pi eu}}pomada _{gama }{fratura {f(z)}{(z-c)^{n+1}}},dz.} The path of integration {gama de estilo de exibição } is counterclockwise around a Jordan curve enclosing c and lying in an annulus A in which {estilo de exibição f(z)} is holomorphic (analytic). The expansion for {estilo de exibição f(z)} will then be valid anywhere inside the annulus. The annulus is shown in red in the figure on the right, along with an example of a suitable path of integration labeled {gama de estilo de exibição } . If we take {gama de estilo de exibição } to be a circle {estilo de exibição |z-c|=varrho } , Onde {estilo de exibição r

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