Abstract.
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.
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Acknowledgments.
The authors are grateful to a referee for his constructive comments which lead to improvement in presentation of this paper and for bringing reference [15] to their attention. Supported in part by the US National Science Foundation under grant nos. 9973427 and 0312113, by the Natural Science Foundation of China under grant no. 10371122 and by the Chinese Academy of Sciences under the program of “One Hundred Distinguished Young Chinese Scientists.”
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Xu, Y., Chen, HL. & Zou, Q. Limit Values of Derivatives of the Cauchy Integrals and Computation of the Logarithmic Potentials. Computing 73, 295–327 (2004). https://doi.org/10.1007/s00607-003-0072-4
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DOI: https://doi.org/10.1007/s00607-003-0072-4