On an asymptotically hyperbolic manifold $(X^{n+1},g)$ , Mazzeo and
Melrose [18] have constructed the meromorphic extension of
the resolvent $R(\lambda):=(\Delta_g-\lambda(n-\lambda))^{-1}$ for
the Laplacian. However, there are special points on
$({1}/{2})(n-\mathbb{N})$ with which they did not deal. We show
that the points of $({n}/{2})-\mathbb{N}$ are at most poles of
finite multiplicity and that the same property holds for the
points of $(({n+1})/{2})-\mathbb{N}$ if and only if the metric is
even. On the other hand, there exist some metrics for which
$R(\lambda)$ has an essential singularity on
$(({n+1})/{2})-\mathbb{N}$ , and these cases are generic. At last,
to illustrate them, we give some examples with a sequence of poles
of $R(\lambda)$ approaching an essential singularity.