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Residue calculus with differential operator. (English) Zbl 0993.32008
If \(u(z)\) is a meromorphic function on a simply connected domain \(X\) of the complex plane, the authors discuss a way of characterizing the principal part of \(u(z)\) via a system of differential operators that annihilates the class \(m\) of \(u(z)\) modulo the sheaf of holomorphic functions as an element of the algebraic local cohomology group (of the sheaf of holomophic functions on \(X\)) with support in the defining ideal \(A\) of \(X\). The system of differential operators is constructed explicitly.
The problem is stated using the duality between the algebraic local cohomology group and the sheaf of holomorphic differential forms. An explicit example is provided. Similar results are said to hold in the higher-dimensional case for Grothendieck residues.

32C30 Integration on analytic sets and spaces, currents
30D30 Meromorphic functions of one complex variable (general theory)
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