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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 a_n 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 be valid anywhere inside this annulus. The annulus is shown in red in the diagram on the right, along with an example of a suitable path of integration labelled γ. In practice, this formula is rarely used because the integrals are difficult to evaluate; instead, one typically pieces together the Laurent series by combining known Taylor expansions. The numbers a_n and c are most commonly taken to be complex numbers, although there are other possibilities