a restricted spacelike surface (e.g. a long thin cylinder, such as an antenna), but different than our previous example, the time dependence of the function is known (e.g. harmonic). This will require the use of a [*Green function*]{} which satisifies: $(\Box - m)\ G({\bf x}) =\delta^4({\bf x})$. When the use of $i$ has been excluded, the inhomogeneous part will take the form, $$G({\bf x}) = \int{ {d^4{\bf k}} \over {(2\pi)^4 [m^2 + {\bf k}^2]}}
[m \cos(k^{\mu}x_{\mu}) + {\bf k} \sin(k^{\mu}x_{\mu})], \eqno(4.6a)$$ $$=(\Box + m)\ \Theta (t)\ \Delta ({\bf x}), \eqno(4.6b)$$ $$=\Theta (t)\ (\Box + m)\ \Delta ({\bf x}), \eqno(4.6c)$$ $$=\Theta (t)\ F({\bf x})\ \basis{4}, \eqno(4.6d)$$ $$=\Theta (t) \int {d^3\vector{P} \over {(2\pi)^3 E}}
\ {\bf p}\ \exp(-\phat \ p^{\mu} x_{\mu}), \eqno(4.6e)$$ where $\Theta(t)$ is a step function, which takes on the value of $+\frac{1}{2}$ for $t>0$, and $-\frac{1}{2}$ for $t<0$. Equation (4.6c) follows from eq. (4.6b) if we note $\delta(t)\Delta({\bf x})=0$. One can go directly from eq. (4.6a) to eq. (4.6e) by doing the contour integral, however the use of $i$ has been excluded. The generator of the residue can possibly be taken to be the geometry associated with $dk^4$, again the (minus) time basis vector $-\basis{4}$ (see for example Hestenes[@Hestenes68]) 