  A set of pupil apodization functions for use with a vortex coronagraph on
telescopes with obscured apertures is presented. We show analytically that
pupil amplitudes given by real-valued Zernike polynomials offer ideal on-axis
starlight cancellation when applied to unobscured circular apertures. The
charge of the vortex phase element must be a nonzero even integer, greater than
the sum of the degree and the absolute value of its azimuthal order of the
Zernike polynomial. Zero-valued lines and points of Zernike polynomials, or
linear combinations thereof, can be matched to obstructions in the pupils of
ground-based telescopes to improve the contrast achieved by a vortex
coronagraph. This approach works well in the presence of a central obscuration
and radial support structures. We analyze the contrast, off-axis throughput,
and post-coronagraph point spread functions of an apodized vortex coronagraph
designed for the European Extremely Large Telescope (E-ELT). This technique
offers very good performance on apertures with large obscuring support
structures similar to those on future 30-40m class ground-based telescopes.
