Goursat's Theorem states that a function complex differentiable at every point of an open subset of the complex plane is in fact holomorphic there, meaning its derivative is itself continuous and the function is infinitely differentiable, without needing to assume continuity of the derivative in advance. Named for Edouard Goursat, it strengthens an earlier version of the Cauchy Integral Theorem that had been proved only under the extra hypothesis of a continuous derivative, showing that hypothesis was never actually necessary.
Facts
StatementGoursat showed that Cauchy's integral theorem can be proven assuming only that the complex derivative exists everywhere in U. 1 Classification
Statement FormCharacterization Theorem 1 Connections
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Principal Result Of
Source Cauchy Integral Theorem (Wikipedia)
Proved By
Source Cauchy Integral Theorem (Wikipedia)
Sources
1. Cauchy's integral theorem (Wikipedia)
DiscussionQuote, Discussion
Cauchy's integral theorem can be proven assuming only that the complex derivative
Cauchy Integral Theorem (Wikipedia)
Wikimedia FoundationPrincipal Result Of: Édouard Goursat, Lead sentence of the article on the result
In mathematics, 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.
Proved By: Édouard Goursat, Article text
Cauchy provided this proof, but it was later proven by Goursat without requiring techniques from vector calculus, or the continuity of partial derivatives.
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