In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities. (More generally, residues can be calculated for any function ${\displaystyle f\colon \mathbb {C} \setminus \{a_{k}\}_{k}\rightarrow \mathbb {C} }$ that is holomorphic except at the discrete points {a_k}_k, even if some of them are essential singularities.) Residues can be computed quite easily and, once known, allow the determination of general contour integrals via the residue theorem.

The residue of a meromorphic function ${\displaystyle f}$ at an isolated singularity ${\displaystyle a}$, often denoted ${\displaystyle \operatorname {Res} (f,a)}$, ${\displaystyle \operatorname {Res} _{a}(f)}$, ${\displaystyle \mathop {\operatorname {Res} } _{z=a}f(z)}$ or ${\displaystyle \mathop {\operatorname {res} } _{z=a}f(z)}$, is the unique value ${\displaystyle R}$ such that ${\displaystyle f(z)-R/(z-a)}$ has an analytic antiderivative in a punctured disk ${\displaystyle 0<\vert z-a\vert <\delta }$.

Alternatively, residues can be calculated by finding Laurent series expansions, and one can define the residue as the coefficient a₋₁ of a Laurent series.

The concept can be used to provide contour integration values of certain contour integral problems considered in the residue theorem. According to the residue theorem, for a meromorphic function ${\displaystyle f}$, the residue at point ${\displaystyle a_{k}}$ is given as:

where ${\displaystyle \gamma }$ is a positively oriented simple closed curve around ${\displaystyle a_{k}}$ and not including any other singularities on or inside the curve.

The definition of a residue can be generalized to arbitrary Riemann surfaces. Suppose ${\displaystyle \omega }$ is a 1-form on a Riemann surface. Let ${\displaystyle \omega }$ be meromorphic at some point ${\displaystyle x}$, so that we may write ${\displaystyle \omega }$ in local coordinates as ${\displaystyle f(z)\;dz}$. Then, the residue of ${\displaystyle \omega }$ at ${\displaystyle x}$ is defined to be the residue of ${\displaystyle f(z)}$ at the point corresponding to ${\displaystyle x}$.

Computing the residue of a monomial

makes most residue computations easy to do. Since path integral computations are homotopy invariant, we will let ${\displaystyle C}$ be the circle with radius ${\displaystyle 1}$ going counter clockwise. Then, using the change of coordinates ${\displaystyle z\to e^{i\theta }}$ we find that