The particular solutions of a two-dimensional infinite wedge for various boundary conditions with weak singularity have been investigated in this paper. The relations of the weak singularities and the discontinuities of the first kind of the boundary variables at a corner of a two-dimensional elastic body have been established. By using the relations, the singular behaviors of the unknown boundary variables at a corner of an elastic body can be obtained before solving the boundary value problem by using the boundary element method (BEM). Especially, if the boundary conditions at a corner are displacements prescribed, the values of the unknown tractions at the corner can be determined in advance. Thus, the difficulty related to the multivalued tractions at a corner in BEM analysis for problems with boundary displacements prescribed has been overcome completely. In addition, more appropriate shape functions for the unknown boundary field variables of a corner element can be constructed, and the accuracy of the BEM may be greatly increased.