Suppose $\Omega$ is a region of $\sR^n$ and $\partial\Omega$ is the boundary of $\Omega$ . Further suppose $f$ is a function $f\colon\partial \Omega\to\sC$ , and suppose $\frac{\partial}{\partial n}$ corresponds to taking a derivative in a direction normal to the boundary $\partial\Omega$ at any point. Then the Neumann problem is to find a function $\phi\colon \Omega\cup \partial \Omega \to\sC$ such that \begin{eqnarray*} \frac{\partial\phi}{\partial n} &=& f,\quad \text{on $\partial \Omega$}, \\ \nabla^2 \phi &=& 0,\quad \text{in $\Omega$}. \end{eqnarray*}Here $\nabla^2$ represents the Laplacian operator and the second condition is that $\phi$ be a harmonic function on $\Omega$ . The condition for the existence of a solution $\phi$ of the Neumann problem is that integral of the normal derivative of the function $\phi$ , calculated over the entire boundary $\partial\Omega$ , vanish. This follows from the identic equation