Laurent series

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]

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).

The Laurent series for a complex function f(z) about a point c is given by

where an and c are constants, with an defined by a line integral that generalizes Cauchy's integral formula:

The path of integration is counterclockwise around a Jordan curve enclosing c and lying in an annulus A in which is holomorphic (analytic). The expansion for 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 . If we take to be a circle , where , this just amounts to computing the complex Fourier coefficients of the restriction of to . The fact that these integrals are unchanged by a deformation of the contour is an immediate consequence of Green's theorem.

One may also obtain the Laurent series for a complex function f(z) at . However, this is the same as when (see the example below).

In practice, the above integral formula may not offer the most practical method for computing the coefficients for a given function ; instead, one often pieces together the Laurent series by combining known Taylor expansions. Because the Laurent expansion of a function is unique whenever it exists, any expression of this form that equals the given function in some annulus must actually be the Laurent expansion of .

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