On the regularization of Cauchy-type integral operators via the density interpolation method and applications

This paper presents a regularization technique for the high order efficient numerical evaluation of nearly singular, principal-value, and finite-part Cauchy-type integral operators. By relying on Cauchy formula, Cauchy-Goursat theorem, on-curve Taylor interpolations input density, proposed methodology allows to recast associated operators as smooth contour integrals. As such, they can be accurately evaluated everywhere in complex plane—including at problematic points near contour—by means elementary quadrature rules. Applications Laplace layer potentials related operators, well computation conformal mappings, are examined detail. The former application, particular, amounts significant improvement over recently introduced harmonic density interpolation method. Spectrally accurate discretization approaches piecewise contours presented. A variety examples, including solution weakly singular hypersingular boundary equations, challenging demonstrate effectiveness accuracy method this context.