  Motivated by mirror symmetry, we study certain integral representations of
solutions to the Gel'fand-Kapranov-Zelevinsky(GKZ) hypergeometric system. Some
of these solutions arise as period integrals for Calabi-Yau manifolds in mirror
symmetry. We prove that for a suitable compactification of the parameter space,
there exists certain special boundary points, which we called maximal
degeneracy points, at which all but one solutions become singular.
