(p_j+{\rm i}\kappa)(p_j^*-{\rm i}\kappa)}\,\tau+\zeta_{j0}+\zeta_{j0}^*,$$ showing that $f$ can be regarded as a function of $\zeta_j\ (j=1, 2, .., N)$. Thus, differentiation of $f$ with respect to $t$ gives $$f_t=\sum_{j=1}^N{\partial f\over\partial\zeta_j}{\partial\zeta_j\over\partial t}=\kappa\rho^2\sum_{j=1}^N\left({1\over p_j}+{1\over p_j^*}\right){\partial f\over\partial\zeta_j}.$$ Similarly, one has $$f_\tau=\sum_{j=1}^N{p_j+p_j^*\over (p_j+{\rm i}\kappa)(p_j^*-{\rm i}\kappa)}{\partial f\over\partial\zeta_j}.$$ The constraints (3.2c) are introduced into the above expression to give $$f_\tau={\kappa\rho^2\over 1+\kappa\rho^2}\sum_{j=1}^N\left({1\over p_j}+{1\over p_j^*}\right){\partial f\over\partial\zeta_j}={1\over 1+\kappa\rho^2\,}f_t.$$ This completes the proof of (3.20a). Repeating the similar procedure, one can show that the relation (3.20b) holds as well. $\Box$

The lemma 3.4 gives the differentiation rules of $f$ and $g$ with respect to $\tau$:

[**Lemma 3.4.**]{}

$$f_\tau={\rm i}|D({\bf z}_\tau^*;{\bf z})|, \eqno(3.21)$$ $$f_{\tau}^*=-{\rm i}|D({\bf z}^*;{\bf z}_\tau)|, \eqno(3.22)$$ $$g_\tau={{\rm i}\over \kappa\rho^2}|D({\bf z}_t^*;{\bf z})|+{1\over \rho^2}|D({\bf z}_t^*;{\bf z}_\tau)|, \eqno(3.23)$$ $$g_{x\tau}={\rm i}|D({\bf z}^*;{\bf z})|+{1\over \rho^2}|D({\bf z}_t^*;{\bf z})|+\kappa |D({\bf z}^*;{\bf z}_\tau)|+{{\rm i}\over \kappa\rho^2}|D({\bf z}_t^*;{\bf z}_x)|
            -{{\rm i \kappa}\over \rho^2}|D({\bf z}_t^*;{\bf z}_\tau)|$$ $$-{1\over \kappa\rho^2}|D({\bf z}_t^*,{\bf z}^*;{\bf z},{\bf z}_x)|-{\kappa\over \rho^2}|D({\bf z}_t^*,{\bf z}^*;{\bf z}_\tau,{\bf z})| 
+{{\rm i}\over \rho^2}|D({\bf z}_t^*,{\bf z}^*;{\bf z}_\tau,{\bf z}_x)|. \eqno(3.24)$$ [**Proof.**]{} If one notes the relations $${\bf z}_{t\tau}=-{{\rm i}\over \kappa}{\bf z}_t+{\rm i}\rho^2{\bf z}_{\tau},\qquad {\bf z}_{x\tau}={\bf z}-{\rm i}\kappa{\bf z}_{\tau},$$ which follows from the definition (3.2a) of $z_j$, the proof can be done straightforwardly along with the same procedure as that used in the proof of lemma 3.1 