  Mellin-Barnes (MB) integrals are well-known objects appearing in many
branches of mathematics and physics, ranging from hypergeometric functions
theory to quantum field theory, solid state physics, asymptotic theory, etc.
Although MB integrals have been studied for more than one century, until now
there is no systematic computational technique of the multiple series
representations of $N$-fold MB integrals for $N>2$. Relying on a simple
geometrical analysis based on conic hulls, we show here a solution to this
important problem. Our method can be applied to resonant (i.e logarithmic) and
nonresonant cases and, depending on the form of the MB integrand, it gives rise
to convergent series representations or diverging asymptotic ones. When
convergent series are obtained the method also allows, in general, the
determination of a single master series for each series representation, which
considerably simplifies convergence studies and/or numerical checks. We
provide, along with this paper, a Mathematica implementation of our technique
with examples of applications. Among them, we present the first evaluation of
the hexagon and double box conformal Feynman integrals with unit propagator
powers.
