# 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]

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:

${\displaystyle a_{n}={\frac {1}{2\pi i}}\oint _{\gamma }{\frac {f(z)}{(z-c)^{n+1}}}\,dz.}$

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<\varrho , 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.

One may also obtain the Laurent series for a complex function f(z) at ${\displaystyle z=\infty }$. However, this is the same as when ${\displaystyle R\rightarrow \infty }$ (see the example below).

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