We give a proof of an integral formula of Berndtsson which is related to the inversion of Fourier-Laplace transforms of $\overline{\partial }$-closed $(n,n-1)$-forms in the complement of a compact convex set in ${ℂ}^n$.

We study the residue current $R^f$ of Bochner-Martinelli type associated with a tuple $f=(f_1,\cdots ,f_m)$ of holomorphic germs at $0\in {\mathbf{C}}^n$, whose common zero set equals the origin. Our main results are a geometric description of $R^f$ in terms of the Rees valuations associated with the ideal $\left(f\right)$ generated by $f$ and a characterization of when the annihilator ideal of $R^f$ equals $\left(f\right)$.