= Coordinate-integral primitive on a polydisc
{title2=$F(z)=\sum_{k=1}^n\int_{a_k}^{z_k}b_k(z_1,\ldots,z_{k-1},t,a_{k+1},\ldots,a_n)\,dt$}
For a closed <holomorphic one-form> $\sum b_kdz_k$ on a polydisc, the displayed coordinate-path integral is a <holomorphic function>. Differentiating in $z_j$ gives the $k=j$ endpoint term; for $k>j$ use $\partial_jb_k=\partial_kb_j$ and integrate that derivative. The resulting sum telescopes to $b_j(z)$. Thus $\partial F=\sum b_kdz_k$, giving an explicit degree-one proof of the <holomorphic Poincaré lemma>.
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