Based on the complex potentials formulation and conformal mapping approach, we derive the fundamental solution for a point force acting in an infinite plane containing an arbitrary shaped notch. The solution satisfies the traction‐free boundary condition along the notch. Therefore, in the boundary element computation of the stress distribution for a notch problem the notch boundary can be excluded from the path of boundary integration, and the high stress gradient around the notch is not disturbed. Several examples are given to show the efficiency and accuracy resulting from the implementation of the present fundamental solution in the boundary element computation of the stress concentrations around notches and the stress intensity factors for elastic crack problems.