Conformal transformation solutions of the extended Motz problem

The Motz problem of 1946 has attracted considerable interest for numerical schemes to accommodate the singularity due to a switch in boundary conditions from Dirichlet to Neumann at a mid-point on one side of a rectangular domain defined by Laplace's equation. Although a detailed solution was provided by means of conformal transformations in 1972 for a harmonic potential function, that solution is well known but largely unrecognised as an exact solution. Moreover, the conformal transformation solutions for its first derivatives, conjugate harmonic function as well as arbitrary position of the boundary singularity point have not been produced. This paper overcomes those shortcomings and provides numerical results for a variety of rectangle side ratios and positions of singularity boundary point.