<math> \frac{\partial^2\psi}{\partial x^2} + \frac{\partial^2\psi}{\partial y^2} + \frac{\partial^2\psi}{\partial z^2} = -f(\mathbf{r}) </math>
When f(<b>r</b>) = 0, one obtains the well-known Laplace equation, which applies to source-free regions.
The second derivative of ψ with respect to the x coordinate can be approximated by the second-order difference:
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In the special case of a uniform grid with Δx = Δy = Δz, it can be shown thatin a source-free region:
<math> \psi(i,j,k) = \frac{1}{6} \big[ \psi(i+1,j,k) + \psi(i-1,j,k) + \psi(i,j+1,k) + \psi(i,j-1,k) + \psi(i,j,k+1) + \psi(i,j,k-1) \big] </math>