Fundamental problems for a weakened infinite plate by a curvilinear hole in a half-plane

Complex variable method (Cauchy integral method) has been applied to derive exact and closed expressions of Goursat functions for the first and second fundamental problems for an infinite plate weakened by a curvilinear hole. The area outside the hole with the hole itself is conformally mapped on the right half-plane by the use of a rational mapping function. This rational mapping consists of complex constants, in order to make the hole take different famous shapes, which can be found throughout the nature. Many previous works are considered as special cases of this work. Also many new cases can be derived from the problem.