  If a differential operator $D$ on a smooth Hermitian vector bundle $S$ over a
compact manifold $M$ is symmetric, it is essentially self-adjoint and so admits
the use of functional calculus. If $D$ is also elliptic, then the Hilbert space
of square integrable sections of $S$ with the canonical left $C(M)$-action and
the operator $\chi(D)$ for $\chi$ a normalizing function is a Fredholm module,
and its $K$-homology class is independent of $\chi$. In this expository
article, we provide a detailed proof of this fact following the outline in the
book "Analytic K-homology" by Higson and Roe.
