  Starting with a mathematical boundary value problem for the magnetic vector
potential in an axisymmetric cylindrical coordinate system, we derive a general
solution for any arbitrary current distribution using the method of Green's
functions. We use this to derive an analytic form for an hourglass magnetic
field pattern created by electrical currents that are concentrated near (but
not confined within) the equatorial plane of a cylindrical coordinate system.
Our solution is not characterized by a cusp at the equatorial plane, as in
previous solutions based on a current sheet. The pattern we derive provides a
very good fit to hourglass magnetic field patterns emerging from
three-dimensional numerical simulations of core formation, and can in principle
be used for source-fitting of observed magnetic hourglass patterns.
