Residue (complex analysis)

Mathematical analysis → Complex analysis
Complex analysis
Complex numbers
Real number Imaginary number Complex plane Complex conjugate Euler's formula
Basic theory
Argument principle Residue Essential singularity Isolated singularity Removable singularity Zeros and poles
Complex functions
Complex-valued function Antiderivative Analytic function Entire function Holomorphic function Meromorphic function Cauchy–Riemann equations Formal power series Laurent series
Theorems
Analyticity of holomorphic functions Cauchy's integral theorem Cauchy's integral formula Residue theorem Liouville's theorem Picard theorem Weierstrass factorization theorem
Geometric function theory
Complex manifold Conformal map Quasiconformal mapping Riemann surface Schwarz lemma Winding number Analytic continuation Riemann's mapping theorem
People
Augustin-Louis Cauchy Leonhard Euler Carl Friedrich Gauss Bernhard Riemann Karl Weierstrass
Mathematics portal
v t e

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.