Let \(U\) be a polydisc in \(\mathbb{C}^n\). Let \(H\) be an analytic hypersurface in \(U\times\mathbb{C}\) such that the projection \(\pi:H\to U\) exhibits \(H\) as a branched analytic covering of \(U\). For \(\varphi\) an \((r,s)\)-form on \(H\), we define \(\text{trace}_\pi(\varphi)\) by \(\langle\text{trace}_\pi(\varphi),\psi\rangle:=\int_H\varphi\wedge\pi^*\psi\), for \(\psi\) an \((n-r,n-s)\)-form of compact support in \(U\). We first show that these courants-traces are locally integrable and that \(\text{trace}_\pi(A^{r,s}_H)\) is spanned over \(C^\infty_U\) by a coherent sheaf which localised over the discriminant of \(\pi\) is equipped with a meromorphic connection with regular singularity. Showing that these courants are invariants under proper modification, we get the exactness of the Dolbault complex of the courants-traces.