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.

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

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

The path of integration ${\displaystyle \gamma }$ is counterclockwise around a closed, rectifiable path containing no self-intersections, 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 ${\displaystyle r<\varrho <R}$, this just amounts to computing the complex Fourier coefficients of the restriction of ${\displaystyle f}$ to ${\displaystyle \gamma }$. The fact that these integrals are unchanged by a deformation of the contour ${\displaystyle \gamma }$ is an immediate consequence of Green's theorem.

...
Wikipedia