Abstract General complex variable boundary element methods are developed for solving potential problems that are reducible to Laplace equations in two dimension. Complex potential nodal equations are derived in a format so that the current computer codes developed for the solution of problems in simply connected domains can be readily extended for the solution of problems in multiply connected domains. The double valued stream functions at the lines of cuts are accounted for by means of perturbation terms in the equations. In order to facilitate development of computer codes, relations are established for the equivalence of nodal points near the lines of the cuts. These relations are given together with coding rules and solution strategies in the paper.