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This paper presents a regularization technique for the high order efficientnumerical evaluation of nearly singular, principal-value, and finite-partCauchy-type integral operators. By relying on the Cauchy formula, theCauchy-Goursat theorem, and on-curve Taylor interpolations of the inputdensity, the proposed methodology allows to recast the Cauchy and associatedintegral operators as smooth contour integrals. As such, they can be accuratelyevaluated everywhere in the complex plane -- including at problematic pointsnear and on the contour -- by means of elementary quadrature rules.Applications of the technique to the evaluation of the Laplace layer potentialsand related integral operators, as well as to the computation conformalmappings, are examined in detail. The former application, in particular,amounts to a significant improvement over the recently introduced harmonicdensity interpolation method. Spectrally accurate discretization approaches forsmooth and piecewise smooth contours are presented. A variety of numericalexamples, including the solution of weakly singular and hypersingular Laplaceboundary integral equations, and the evaluation of challenging conformalmappings, demonstrate the effectiveness and accuracy of the densityinterpolation method in this context.

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