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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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 FoundationProved By: Édouard Goursat, Article textQuote, 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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