Given a compact Riemannian manifold (M^d, g), a finite dimensional representationρ:π₁(M)→GL(V) of the fundamental groupπ₁(M) on a vector spaceVof dimensionland a Hermitian structureμon the flat vector bundle[formula]associated toρ, Ray–Singer [RS] have introduced the analytic torsionT=T(M, ρ, g, μ)>0. Witten's deformationd_q(t) of the exterior derivatived_q,d_q(t)=e^−htd_qe^ht, withh:M→Ra smooth Morse function, can be used to define a deformationT(h, t)>0 of the analytic torsionTwithT(h, 0)=T. The main results of this paper are to provide, assuming that grad_ghis Morse Smale, an asymptotic expansion for logT(h, t) fort→∞ of the form[formula]and to present two different formulae fora₀. As an application we obtain a shorter derivation of results due to Ray–Singer [RS], Cheeger [Ch], Müller [Mu1, 2] which, in increasing generality, concern the equality for odd dimensional manifolds of the analytic torsion with the average of the Reidemeister torsion corresponding to the triangulation T =(h, g) and the dual triangulation T _D =(d−h, g).