On radial limit functions for entire solutions of second order elliptic equations in $\mathbf{R}^2$

Given a homogeneous elliptic partial differential operator $L$ of order two with constant complex coefficients in $\bold R^2$, we consider entire solutions of the equation $Lu=0$ for which
$$
\lim_{r \rightarrow \infty} u (re^{i\varphi}) =: U (e^{i\varphi})
$$
exists for all $\varphi \in [0, 2\pi)$ as a finite limit in $\bold C$. We characterize the possible "radial limit functions" $U$. This is an analog of the work of A. Roth for entire holomorphic functions. The results seem new even for harmonic functions.