Limit Values of Derivatives of the Cauchy Integrals and Computation of the Logarithmic Potentials

We derive limit values of high-order derivatives of the Cauchy integrals, which are extensions of the Plemelj-Sokhotskyi formula. We then use them to develop the Taylor expansion of the logarithmic potentials at the normal direction. Based on the Taylor expansion and numerical integration methods for weekly singular functions using grid points, we design fast algorithms for computing the logarithmic potentials. We prove that these methods have an optimal order of convergence with a linear computational complexity. Numerical examples are included to confirm the theoretical estimates for the methods.