  We compute the gap probability that a circle of radius r around the origin
contains exactly k complex eigenvalues. Four different ensembles of random
matrices are considered: the Ginibre ensembles and their chiral complex
counterparts, with both complex (beta=2) or quaternion real (beta=4) matrix
elements. For general non-Gaussian weights we give a Fredholm determinant or
Pfaffian representation respectively, depending on the non-Hermiticity
parameter. At maximal non-Hermiticity, that is for rotationally invariant
weights, the product of Fredholm eigenvalues for beta=4 follows from beta=2 by
skipping every second factor, in contrast to the known relation for Hermitian
ensembles. On additionally choosing Gaussian weights we give new explicit
expressions for the Fredholm eigenvalues in the chiral case, in terms of
Bessel-K and incomplete Bessel-I functions. This compares to known results for
the Ginibre ensembles in terms of incomplete exponentials. Furthermore we
present an asymptotic expansion of the logarithm of the gap probability for
large argument r at large N in all four ensembles, up to including the third
order linear term. We can provide strict upper and lower bounds and present
numerical evidence for its conjectured values, depending on the number of exact
zero eigenvalues in the chiral ensembles. For the Ginibre ensemble at beta=2
exact results were previously derived by Forrester.
