  From incompressible flows to electrostatics, harmonic functions can provide
solutions to many two-dimensional problems and, similarly, the director field
of a planar nematic can be determined using complex analysis. We derive a
closed-form solution for a quasi-steady state director field induced by an
arbitrarily large set of point defects and circular inclusions with or without
fixed rotational degrees of freedom, and compute the forces and torques acting
on each defect or inclusion. We show that a complete solution must include two
types of singularities, generating a defect winding number and its spiral
texture, which have a direct effect on defect equilibrium textures and their
dynamics. The solution accounts for discrete degeneracy of topologically
distinct free energy minima which can be obtained by defect braiding. The
derived formalism can be readily applied to equilibrium and slowly evolving
nematic textures for active or passive fluids with multiple defects present
within the orientational order.
