Coordinate-integral primitive on a polydisc ($F(z)={\sum }_{k=1}^n{\int }_{a_k}^{z_k}{b}_k(z_1,\dots ,z_{k-1},t,a_{k+1},\dots ,a_n)dt$)

For a closed holomorphic one-form $\sum b_{k}d{z}_k$ 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 ${\partial }_j{b}_k={\partial }_k{b}_j$ and integrate that derivative. The resulting sum telescopes to $b_j(z)$. Thus $\partial F=\sum b_{k}d{z}_k$, giving an explicit degree-one proof of the holomorphic Poincaré lemma.