  We prove that the resolvent of a linear operator pencil is analytic on an
open annulus if and only if the coefficients of the Laurent series satisfy a
system of fundamental equations and are geometrically bounded. Our analysis
extends earlier work on the fundamental equations to include the case where the
resolvent has an isolated essential singularity. We find a closed form for the
resolvent and use the fundamental equations to establish key spectral
separation properties when the resolvent has only a finite number of isolated
singularities. Finally we show that our results can also be applied to
polynomial pencils.
