  Following the work of Conrey, Rubinstein and Snaith and Forrester and Witte
we examine a mixed moment of the characteristic polynomial and its derivative
for matrices from the unitary group U(N) (also known as the CUE) and relate the
moment to the solution of a Painleve differential equation. We also calculate a
simple form for the asymptotic behaviour of moments of logarithmic derivatives
of these characteristic polynomials evaluated near the unit circle.
