In mathematics, the Laurent series of a complex function ${\displaystyle f(z)}$ 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 had previously described it in a paper written in 1841 but not published until 1894.

The Laurent series for a complex function ${\displaystyle f(z)}$ about an arbitrary point ${\displaystyle c}$ is given by $${\displaystyle f(z)=\sum _{n=-\infty }^{\infty }a_{n}(z-c)^{n},}$$ where the coefficients ${\displaystyle a_{n}}$ are defined by a contour 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 ${\displaystyle c}$ and lying in an annulus ${\displaystyle A}$ 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 ${\displaystyle \gamma }$. When ${\displaystyle \gamma }$ is defined as the circle ${\displaystyle |z-c|=\varrho }$, where ${\displaystyle r<\varrho <R}$, this 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 ${\displaystyle f(z)}$ at ${\displaystyle z=\infty }$. However, this is the same as when ${\displaystyle R\rightarrow \infty }$.

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 ${\displaystyle f(z)}$.

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

Consider for instance the function ${\displaystyle f(x)=e^{-1/x^{2}}}$ with ${\displaystyle f(0)=0}$. As a real function, it is infinitely differentiable everywhere; as a complex function however it is not differentiable at ${\displaystyle x=0}$. The Laurent series of ${\displaystyle f(x)}$ is obtained via the power series representation, $${\displaystyle e^{-1/x^{2}}=\sum _{n=0}^{\infty }(-1)^{n}\,{x^{-2n} \over n!},}$$ which converges to ${\displaystyle f(x)}$ for all ${\displaystyle x\in \mathbb {C} }$ except at the singularity ${\displaystyle x=0}$. The graph on the right shows ${\displaystyle f(x)}$ in black and its Laurent approximations $${\displaystyle \sum _{n=0}^{N}(-1)^{n}\,{x^{-2n} \over n!},\quad \forall N\in \mathbb {N} ^{+}.}$$ As ${\displaystyle N\to \infty }$, the approximation becomes exact for all (complex) numbers ${\displaystyle x}$ except at the singularity ${\displaystyle x=0}$.

More generally, Laurent series can be used to express holomorphic functions defined on an annulus, much as power series are used to express holomorphic functions defined on a disc.