Determining of an analytic function on its analytic domain by Cauchy-Riemann equation with special kind of boundary conditions (Q1884512)

The authors investigate the solvability of the Cauchy-Riemann system in the upper half plane for functions \(u(x_1,x_2)\) tending to 0 for \(x_1\to\pm\infty\), \(x_2\to\infty\). Application of the Fourier and the Laplace transform gives explicit solution formulas if the problem is solvable, since a Dirichlet boundary condition on the real axis cannot be prescribed arbitrarily. As a third method, fundamental solutions are introduced and the solution is given provided the difference \(u(x_1,0)-u(-x_1,0)\) is a prescribed function. Unfortunately, the authors do not use either Cauchy's nor Poisson's integral formula, which would be better suited to the problem.