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Residue (complex analysis) - Wikipedia

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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 𝑓 : 𝐶 ∖ { 𝑎 𝑘 } 𝑘 → 𝐶 that is holomorphic except at the discrete points {ak}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 𝑓 at an isolated singularity 𝑎 , often denoted Res ⁡ ( 𝑓 , 𝑎 ) , Res 𝑎 ⁡ ( 𝑓 ) , Res 𝑧 = 𝑎 ⁡ 𝑓 ( 𝑧 ) or res 𝑧 = 𝑎 ⁡ 𝑓 ( 𝑧 ) , is the unique value 𝑅 such that 𝑓 ( 𝑧 ) − 𝑅 / ( 𝑧 − 𝑎 ) has an analytic antiderivative in a punctured disk 0 < | 𝑧 − 𝑎 | < 𝛿 . Alternatively, residues can be calculated by finding Laurent series expansions, and one can define the residue as the coefficient a−1 of a Lauren

Residue (complex analysis) - Wikipedia Jump to content From Wikipedia, the free encyclopedia Attribute of a mathematical function "Residue (mathematics)" redirects here. For the concept in modular arithmetic, see Residue number system . 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

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