  In this paper we study the asymptotic behavior (in the sense of meromorphic
functions) of the zeta function of a Laplace-type operator on a closed manifold
when the underlying manifold is stretched in the direction normal to a dividing
hypersurface, separating the manifold into two manifolds with infinite
cylindrical ends. We also study the related problem on a manifold with boundary
as the manifold is stretched in the direction normal to its boundary, forming a
manifold with an infinite cylindrical end. Such singular deformations fall
under the category of "analytic surgery", developed originally by Hassell,
Mazzeo and Melrose \cite{mazz95-5-14,hass95-3-115,hass98-6-255} in the context
of eta invariants and determinants.
