Residue (mathematics)

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In complex analysis, the residue of a function f holomorphic in an open set ${\displaystyle \Omega }$ with possible exception of a point ${\displaystyle z_{0}\in \Omega }$ where the function may admit an isolated singularity, is a particular number describing behaviour of f around ${\displaystyle z_{0}}$.

More precisely, if a function f is holomorphic in a neighbourhood of ${\displaystyle z_{0}}$ (but not necessarily at ${\displaystyle z_{0}}$ itself), with either a removable singularity or a pole at ${\displaystyle z_{0}}$, then it can be represented as a Laurent series around this point, that is

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

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

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

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.