Laurent series

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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 1841 but did not publish it at the time.

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

where the an are constants, defined by a line integral which is a generalization of Cauchy's integral formula:

The path of integration γ is counterclockwise around a closed, rectifiable path containing no self-intersections, enclosing c and lying in an annulus A in which f(z) is holomorphic (analytic). The expansion for f(z) will then be valid anywhere inside the annulus. The annulus is shown in red in the diagram on the right, along with an example of a suitable path of integration labeled γ. If we take γ to be a circle  |z-c| = \varrho, where r < \varrho < R, this just amounts to computing the complex Fourier coefficients of the restriction of f to γ. The fact that these integrals are unchanged by a deformation of the contour γ is an immediate consequence of Stokes' theorem.

In practice, the above integral formula may not offer the most practical method for computing the coefficients an for a given function f(z); 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 actually equals the given function f(z) in some annulus must actually be the Laurent expansion of f(z).

Contents

Convergent Laurent series

Laurent series with complex coefficients are an important tool in complex analysis, especially to investigate the behavior of functions near singularities.

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