Residue (mathematics)

In complex analysis, the residue of a complex function f holomorphic in a neighbourhood $\Omega$ of a point $z_{0}\in \mathbb {C}$ is a particular number characterising behaviour of f around this point.

More formally, if a function f is holomorphic in a neighbourhood of $z_{0}$ then it can be represented as the Laurent series around this point, that is

$f(z)=\sum _{n=-N}^{\infty }c_{n}(z-z_{0})^{n}$

with some $N\in \mathbb {N}$ and coefficients $c_{n}\in \mathbb {C} .$

The coefficient $c_{-1}$ is the residue of f at $z_{0}$, denoted as $\mathrm {Res} (f,z_{0})$ or $\displaystyle \mathop {\mathrm {Res} } _{z=z_{0}}f.$

Although the choice of the coefficient may look arbitrary, it turns out that it is well motivated by the particularly important role played by this number in the theory of complex functions. For example, the residue allows to evaluate path integrals of the function f via the residue theorem. This technique finds many applications in real analysis as well.