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 {displaystyle f(z)=sum _{n=-infty }^{infty }a_{n}(z-c)^{n},} where an and c are constants, with an defined by a line integral that generalizes Cauchy's integral formula: {displaystyle a_{n}={frac {1}{2pi i}}oint _{gamma }{frac {f(z)}{(z-c)^{n+1}}},dz.} The path of integration {displaystyle gamma } is counterclockwise around a Jordan curve enclosing c and lying in an annulus A in which {displaystyle f(z)} is holomorphic (analytic). The expansion for {displaystyle 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 {displaystyle gamma } . If we take {displaystyle gamma } to be a circle {displaystyle |z-c|=varrho } , where {displaystyle r

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