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\end{equation}  In the reversedo  we have \begin{equation} e^{\Psi(x,t)} = \prod_n e^{a_n\psi_n(x,t)}  \end{equation}  If $\Psi$ and $\psi$ are complex valued, we then have a seperation of parts \begin{equation}  \Psi(x,t) = R(x,t) +i\Upsilon(x,t) \\  \psi(x,t) = r(x,t) +iu(x,t) \\  e^{R(x,t)}(\mathrm{cos}(\Upsilon(x,t))+i\mathrm{sin}(\Upsilon(x,t)) = \prod_n e^{a_nr_n(x,t)} (\mathrm{cos}(a_nu_n(x,t))+i\mathrm{sin}(a_nu_n(x,t))  \end{equation}