  We develop a method for calculating the electron-phonon vertex in polar
semiconductors and insulators from first principles. The present formalism
generalizes the Fr\"ohlich vertex to the case of anisotropic materials and
multiple phonon branches, and can be used either as a post-processing
correction to standard electron-phonon calculations, or in conjunction with
{\it ab initio} interpolation based on maximally localized Wannier functions.
We demonstrate this formalism by investigating the electron-phonon interactions
in anatase TiO$_2$, and show that the polar vertex significantly reduces the
electron lifetimes and enhances the anisotropy of the coupling. The present
work enables {\it ab initio} calculations of carrier mobilities, lifetimes,
mass enhancement, and pairing in polar materials.
