  I calculate the anomalous dimension governing the Q^2 evolution of the gluon
(and structure functions) coming from the running coupling BFKL equation. This
may be expressed in an exact analytic form, up to a small ultraviolet
renormalon contribution, and hence the corresponding splitting function may be
determined precisely. Rather surprisingly it is most efficient to expand the
gluon distribution in powers of alpha_s(Q^2) rather than use the traditional
expansion where all orders of alpha_s\ln(1/x) are kept on an equal footing. The
anomalous dimension is very different from that obtained from the fixed
coupling equation, and leads to a powerlike behaviour for the splitting
function as x ->0 which is far weaker, i.e. about x^(-0.2). The NLO corrections
to the anomalous dimension are rather small, unlike the fixed coupling case,
and a stable perturbative expansion is obtained.
