On solvability of linear systems generated by the complex variable boundary element method

Within the complex variable boundary element method, an approximate solution is determined by a Cauchy-type integral whose density is a piecewise linear function. Such an integral can be expressed by a linear combination of some functions that can be chosen in many ways. The choice influences properties of a linear system that arises by discretization of some boundary value problem. One choice is presented that allows to deduce some results about the system solvability. It is demonstrated on the Dirichlet problem.