We discuss analytic regularization methods used to obtain the renormalized vacuum energy of quantum fields in an arbitrary ultrastatic spacetime. After proving that the $\ensuremath{\zeta}$-function method is equivalent to the cutoff method with the subtraction of the polar terms, we present two examples where the analytic extension method gives a finite result, but in disagreement with the cutoff and $\ensuremath{\zeta}$-function methods.