The definition of the potential related quantities in (15) and (16) capturing the Cauchy Principal value of the surface integrals are
CPV], the other potential magnitude in (27), denotes the contribution of the n-th div-TO basis function to the Cauchy Principal value of the gradient of the electric scalar potential.
Sine methods, Hilbert transform, Cauchy principal value
The a's and b's may be complex (with the b's not equal to zero), but the integral is assumed to be well defined, possibly as a Cauchy principal value.
J] is negative, the Cauchy principal value is given by
Calculations were performed with the package MATHEMATICA, matrices D(x, y) and P(x, y) were computed symbolically, the approximate solutions of the boundary integral equations were obtained by means of cubic splines, and the Cauchy principal values
were computed with Gaussian quadrature.