| Abstract: | An exact expression is derived for the general finite-part integral  over an inclined ellipticaldomain Ω. r denotes the distance of a point in Ω to the singular point $\left({x,y} \right).f = x_{^0 }^i y_0^j \sqrt {Z\left({x_{0,} y_0 }\right)}$ is a general function of the Cartesian co-ordinates x0,y0. The boundary of the region Ω represents the equation Z(x0, y0)=O. These integrals appear during the numerical solution of plane crack problems in three-dimensional elasticity where they are the dominant part of a hypersingular integral equation. The availability of exact expressions for the integrals with arbitrary integers i and j will increase the accuracy of the numerical results and, simultaneously, lead to quicker numerical results. The considered finite-part integral can be expressed in closed form as function of complete elliptical integrals or Gauss hypergeometric functions, respectively. Formuias for special cases and some i, j values and their numerical verification are given in Appendices II and III. |
