  The Wilson-Fisher fixed point with $O(N)$ universality in three dimensions is
studied using the renormalisation group. It is shown how a combination of
analytical and numerical techniques determine global fixed point solutions to
leading order in the derivative expansion for real or purely imaginary fields
with moderate numerical effort. Universal and non-universal quantitites such as
scaling exponents and mass ratios are computed, for all $N$, together with
local fixed point coordinates, radii of convergence, and parameters which
control the asymptotic behaviour of the effective action. We also explain when
and why finite-$N$ results do not converge pointwise towards the exact
infinite-$N$ limit. In the regime of purely imaginary fields, a new link
between singularities of fixed point effective actions and singularities of
their counterparts by Polchinski are established. Implications for other
theories are indicated.
