Reflection principle for solutions of the Helmholtz equation in a half-space

In this paper we consider the reflection principle for functions satisfying the Helmholtz equation in a half-space. The idea of a proof is to perform continuation of a solution of the Helmholtz equation from an upper half-space into the entire space. It is obtained an analog of Liuvill' theorem. This result states that the function which satisfies the Helmholtz equation (with negative parameter) in the upper half-space, which grows not faster than the power function in the upper half-space and which is equal to zero on a hyperplane is identically equal to zero in the entire space.