Summary: A new method is presented for the exact solution in closed form of a mixed boundary value problem of potential theory when the potential is prescribed on one half-plane and the charge-density distribution is prescribed on the other half-plane. The method is based on a new integral representation for the reciprocal of the distance between two points. Its substitution into the simple layer distribution leads to an integral equation which can be solved exactly. The general results are applied to solving a punch problem and a half-plane crack problem. A complete solution for the fields of stresses and displacements is given in closed form and in terms of elementary functions.