  A Gaussian fluctuation formula is proved for linear statistics of complex
random matrices in the case that the statistic is rotationally invariant. For a
general linear statistic without this symmetry, Coulomb gas theory is used to
predict that the distribution will again be a Gaussian, with a specific mean
and variance. The variance splits naturally into a bulk and surface
contibution, the latter resulting from the long range correlations at the
boundary of the support of the eigenvalue density.
