For several elasticity problems, solution representations for the displacements and stresses are available. The solution representations are given in terms of “arbitrary” complex valued functions. For any choice of the complex functions, the governing differential equations are automatically satisfied. Complex solution representations are therefore useful for applications of the Trefftz method. For the analysis of local stress concentrations, due to the local geometry of the boundary curve, such solution representations can be very helpful in the construction of appropriate series of Trefftz functions. In this paper, a few examples are given to demonstrate how to construct Trefftz functions for special purpose finite elements, which include the local solution behavior around a stress concentration or stress singularity.