Is there a Cauchy integral formula for holomorphic functions on complex manifolds?
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1$\begingroup$ Could this be a duplicate of MO261860? A short answer is that, if you can locally represent the location of the pole hypersurface as $z_1=0, z_2=0, \ldots$, you simultaneously use the residue for each of the complex coordinates $z_1,z_2,\ldots$, with Cauchy being a special case. $\endgroup$– Igor KhavkineMay 13, 2023 at 14:10
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