  We show that the classical kernel and domain functions associated to an
n-connected domain in the plane are all given by rational combinations of three
or fewer holomorphic functions of one complex variable. We characterize those
domains for which the classical functions are given by rational combinations of
only two or fewer functions of one complex variable. Such domains turn out to
have the property that their classical domain functions all extend to be
meromorphic functions on a compact Riemann surface, and this condition will be
shown to be equivalent to the condition that an Ahlfors map and its derivative
are algebraically dependent. We also show how many of these results can be
generalized to finite Riemann surfaces.
