I've been struggling these last couple of days to see the connection, if at all there is one, between the following facts:

  1. For holomorphic functions $f$, $\mathrm{d}(f(z)\mathrm{d}z) = 0$.
  2. In a simply connected domain, a holomorphic function has a primitive, i.e. there exists a function $g$ defined over this domain such that $g'=f$.
  3. The de Rham Cohomology "measures the failure of closed forms to be exact".

Trying to convince myself that Cauchy's theorem is somehow geometrically intuitive led me to the one-line proof where point 1 above is combined with Stoke's theorem, which in turn has led me to wonder if there was something going on at the level of differential forms over $\mathbb{C}$. I apologise if the question is unclear; as I said, I have the feeling like there's a revelation about holomorphic functions dancing just out of my reach.


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