This long chapter is devoted to recalling the needed analytic tools. It contains preliminaries on pseudo-differential operators, acting on several function spaces on compact manifolds. Analytic results on non-compact manifolds of bounded geometry are also recalled. Quite a few notions from small b-calculus are also needed, with a special focus on supported and extendible distributions, as well as distributions that are conormal and dual-conormal at the boundary or another submanifold. Special emphasis is placed on the natural topology of these distribution subspaces and on the short exact sequences defined by taking supports at the boundary or submanifold. Next, we consider Witten’s deformation of the de Rham operator acting on differential forms or currents. The local index formula and local trace formula are stated for Witten’s perturbation. The main tool recalled in this part is a tempered distribution associated to some closed 1-forms.