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?