The integral of a holomorphic function around any closed contour lying, together with its interior, inside a simply connected domain of holomorphy is zero. Proved by Augustin-Louis Cauchy, it is the foundation on which the rest of complex integration theory is built.
Facts
StatementIf f is holomorphic throughout a simply connected domain, then for any simple closed contour lying in that domain, the contour integral of f around it is zero. 1 Classification
Statement Form Connections
Sources
1. Cauchy Integral Theorem (Wikipedia)
Wikimedia FoundationLead section, opening sentenceQuote, Lead section, opening sentence
the Cauchy integral theorem (also known as the Cauchy-Goursat theorem) in complex analysis, named after Augustin-Louis Cauchy (and Édouard Goursat), is an important statement about line integrals for holomorphic functions in the complex plane.
Cauchy Integral Theorem (MathWorld)
Theorem statementQuote, Theorem statement
if f(z) is analytic in some simply connected region R, then ∮_γ f(z)dz=0 for any closed contour γ completely contained in R.
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