The integral of a meromorphic function around a closed contour equals two pi i times the sum of the residues of the function's poles enclosed by the contour. A powerful generalization of the Cauchy integral theorem, it is a standard tool for evaluating real integrals via complex analysis.
Facts
StatementThe residue theorem, also called Cauchy's residue theorem, states that a contour integral of an analytic function around a closed curve can be evaluated as the sum of the residues at the poles enclosed by that curve. 1 Classification
Statement Form Connections
Sources
1. Residue Theorem (Wikipedia)
Wikimedia FoundationLead section, opening sentenceQuote, Lead section, opening sentence
In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves
2. Augustin-Louis Cauchy, Biography (MacTutor History of Mathematics)
MacTutor History of Mathematics Archive, University of St AndrewsBiography narrative, on Cauchy's calculus of residuesQuote, Biography narrative, on Cauchy's calculus of residues
He began a study of the calculus of residues in 1826 in Sur un nouveau genre de calcul analogue au calcul infinitesimal
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