  The success of the moving puncture method for the numerical simulation of
black hole systems can be partially explained by the properties of stationary
solutions of the 1+log coordinate condition. We compute stationary 1+log slices
of the Schwarzschild spacetime in isotropic coordinates in order to investigate
the coordinate singularity that the numerical methods have to handle at the
puncture. We present an alternative integration method to obtain isotropic
coordinates that simplifies numerical integration and that gives direct access
to a local expansion in the isotropic radius near the puncture. Numerical
results have shown that certain quantities are well approximated by a function
linear in the isotropic radius near the puncture, while here we show that in
some cases the isotropic radius appears with an exponent that is close to but
unequal to one.
