  In this paper, we consider $\alpha$-harmonic functions in the half space
$\mathbb{R}^n_+$: \begin{equation} \left\{\begin{array}{ll}
(-\Delta)^{\alpha/2} u(x)=0,~u(x)>0, & x\in\mathbb{R}^n_+, \\ u(x)\equiv 0, &
x\notin \mathbb{R}^{n}_{+}. \end{array}\right. \end{equation} We prove that all
the solutions have to assume the form
  \begin{equation} u(x)=\left\{\begin{array}{ll}Cx_n^{\alpha/2}, & \qquad
x\in\mathbb{R}^n_+, \\ 0, & \qquad x\notin\mathbb{R}^{n}_{+},
\end{array}\right. \label{2} \end{equation} for some positive constant $C$.
