Formal expansion at a smooth point (${\hat{\mathcal{O}}}_{X,x}\cong k[[T_1,\dots ,T_d]]$)

A chosen regular system of parameters gives the displayed completion isomorphism by sending $T_i$ to $t_i$. The associated graded ring of a regular local ring is the polynomial ring on their initial forms. Successively removing the homogeneous initial form of each remainder gives a unique formal series for every local function. The Krull intersection theorem embeds the local ring in its completion. The power-series characterization is also given in Stacks Project Lemma 10.160.10.