  The monodromy of torus bundles associated to completely integrable systems
can be computed using geometric techniques (constructing homology cycles) or
analytic arguments (computing discontinuities of abelian integrals). In this
article we give a general approach to the computation of monodromy that
resembles the analytical one, reducing the problem to the computation of
residues of polar 1-forms. We apply our technique to three celebrated examples
of systems with monodromy (the champagne bottle, the spherical pendulum, the
hydrogen atom) and to the case of non degenerate focus-focus singularities,
re-obtaining the classical results. An advantage of this approach is that the
residue-like formula can be shown to be local in a neighborhood of a
singularity, hence allowing the definition of monodromy also in the case of
non-compact fibers. This idea has been introduced in the literature under the
name of scattering monodromy. We prove the coincidence of the two definitions
with the monodromy of an appropriately chosen compactification.
