Since the initial electric field is polarizing the material, the polarization vector must have the same space and time dependence P ( r , t ) = P 0 e i ( k z − − ω ω t ) . More detail about this assumption is discussed by Wangsness. Plugging this into the integral and expressing in terms of Cartesian coordinates produces π π e , p ( r , t ) = P 0 e i ( k z − − ω ω t ) 4 π π ϵ ϵ 0 ∫ ∫ 0 ∞ ∞ d z ′ e i k ( z ′ − − z ) ∫ ∫ − − ∞ ∞ ∞ ∞ d x ′ ∫ ∫ − − ∞ ∞ ∞ ∞ d y ′ e i k 0 | r − − r ′ | | r − − r ′ | .