Completely explicit solutions of Eshelby's problems of smooth inclusions embedded in a circular disk, full- and half-planes

The conventional Eshelby's problems of smooth inclusions in two-dimensional space are touched in this paper. When the smooth inclusion is characterized by the Laurent polynomial, using the solution of the full-plane as basis, the solutions of a finite domain can be decomposed into a basic part and an auxiliary part. The K-M potentials, including the basic and auxiliary parts, of the circular disk, full- and half-plane problems are explicitly solved in the form of polynomials. The coefficients of the polynomials are determined by the geometric parameters characterizing the smooth inclusion and its position. The solutions for elliptical and truncated pentagram inclusions are used as examples to show the correctness and validity of the presented solutions.