A complex variable boundary element method for principal value integrals

The trapezoid rule is often used to evaluate the singular integral equation for the induced velocity of vortex sheets. This quadrature, or Gaussian quadrature more generally, results in an induced velocity field that is meromorphic, i.e. analytic everywhere except at a discrete subset of isolated points. More plainly, such quadrature rules are equivalent to a set of point singularities. On the other hand, the velocity field induced by a continuous vortex sheet is sectionally holomorphic, meaning a discontinuity (in tangential velocity) exists at any point comprising the sheet. We take an alternative approach that aims to preserve this topological feature of the sheet. The velocity discontinuity is effected by the branch cuts of complex logarithms that appear in the quadrature expression. Using complex variables allows a proper treatment of the local contribution to the principal value integral.