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 the linearization of the current function in the axial region, we find an 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, backward particle flow at small polar angles can be excluded and the stationary cascade scenario looks admissible. The present paper is the first step towards a complete analytic description of the pulsar force-free magnetosphere allowing for plasma-producing gaps and pulsar current circuit closure.