Abstract: In engineering fracture analysis,random boundary imperfections as notches or cracks are common and important.In this paper,we have worked out a class of methods for solving two-dimensional problems of random boundary notches and/or cracks by an extension of Muskhelishvili's method.Through successive applications of analytic conti-nuation,Laurent series expansion and conformal mapping,we finally arrived at a set of linear algebraic equations governing the problem,Then standard com puting program of linear algebraic eq nations may he used to com plete the solution.It is noticed that the prescn method may give solutions for the cases in which there are certain smooth load distrihutmns on the notch surfaces.