# Extension of harmonic function at infinity

Can a harmonic function defined on the upper half-plain (or any domain which is unbounded) be extended to the point at infinity. If so, under what condition. What happens to the mean value property then ? Do we still get a integral representation of some sort. Please suggest a reference.

Thank you.

Of course these theorems need a growth condition. For that book for example $|F(z)| \leq \frac{M}{im(z)}$ for some constant $M > 0$. But I think one can extend this to some moderate growth at infinity. For example $|F(z)| = O(\sqrt{|z|})$ as $z\to\infty$ is ok.