  The stationary axisymmetric force-free magnetosphere of a pulsar is studied
analytically. The pulsar equation is solved in the region close to the magnetic
axis. Proceeding from linearization of the current function in the axial
region, we find the axial magnetic flux function valid at any altitude above
the neutron star. This function is used as a starting approximation to develop
series for the non-linear pulsar equation in the polar region. Taking into
account the quasi-monopolar character of the pulsar magnetic flux at infinity,
we obtain unique asymptotic series for the flux and current functions. At
infinity, both functions are close but not equivalent to those known for the
case of a force-free monopole. The flux function at the top of the polar gap is
found to differ from the dipolar one at the neutron star surface. With our
results, the transverse current sheet closing the pulsar circuit at the neutron
star surface is consistently incorporated into the global magnetospheric
structure, the backward particle flow at small polar angles can be excluded and
the stationary cascade scenario looks admissible. The present paper is the
first step toward complete analytic description of the pulsar force-free
magnetosphere allowing for the plasma-producing gaps and pulsar current circuit
closure.
