A numerical method based on the generalized Shannon–Whittaker representation theorem for solving Helmholtz equation

In this article, we first apply the contour integral method to generalize the Shannon–Whittaker theorem to the case for the multi-valued analytic functions. Based on this result we obtain the numerical solution for the Helmholtz equation. In order to overcome the difficulty that the coerciveness does not hold, we prove the existence and uniqueness of the solution to Helmholtz equation with the third boundary condition in the upper half plane.