I'm studying complex analysis and am curious about its history.

Did Cauchy know that holomorphic functions (to have derivative in every point of an open set) are infinitely differentiable? And that they are analytic? (admit a power series expansion at each point)?

If not, who was the first to prove these things? Did Goursat use his theorem (the integral of a function holomorphic inside a triangle is zero over the triangle border, even if we don't use green theorem, that is we don't need the derivative to be continuous) to prove that holomorphism implies analyticity?

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    $\begingroup$ "Know" is a tough word to interpret when it comes to Math in Cauchy's time because a real number had not yet been rigorously defined. Cauchy talked about sequences that, roughly translated, "converged on themselves," which is what we now call a Cauchy sequence. Cauchy died in 1857, well before Cantor's 1871 rigorous construction of a real number. Goursat died in 1936, and he lived during a time when a rigorous demonstration made sense, when real/complex numbers had been made rigorous, and when compactness was known. $\endgroup$ May 1, 2016 at 22:50
  • $\begingroup$ Maybe relevant: zulfahmed.files.wordpress.com/2018/07/…. $\endgroup$ Apr 12, 2019 at 14:58


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