Put $\eta = r+it, r>0$ and $K_{\eta}(z, w, \lambda)= \frac {\mid \lambda\mid }{2\sinh(\eta \mid \lambda \mid) } e^{\frac{\mid\lambda\mid \coth(\eta  \lambda ) \mid zw \mid ^{2}}{4}} e^{i \frac{\lambda}{2} Im(z\cdot \bar{w})}$. and the Fourier transform in the last variable of the function $K_{\eta}(z, w, \lambda)$ is $\hat {K_{\eta}(z, w)}(\tau) = \int_{\mathbb R} K_{\eta}(z, w, \lambda) e^{i\lambda \tau} d\lambda $, ($z, w \in \mathbb C^{n}, \lambda \in \mathbb R $). My question is: can we evaluate this integral ($\hat {K_{\eta}(z, w)}(\tau)$), or can we say it is convergent, or can we get just estimate; or would you like suggest some method to handle such oscillatory integral,?
Let me give a rather general answer, hopefully helping you to cope with your case. We consider $$ I(\lambda)=\int_{\mathbb R^d} e^{i\lambda\phi(x)}a(x,\lambda) dx, $$ where $\lambda \ge 1$ is a real parameter (going to $+\infty$), $\phi$ is the phase, a smooth complexvalued function such that $\Im \phi\ge 0,$ $$ \quad d\Re \phi=0 \text{ and } \Im\phi=0 \Longrightarrow \det(\phi'')\not=0. \tag{$\natural$} $$ The amplitude $a$ is in the Schwartz class (wrt the $x$ variable). Then $I(\lambda)=O(\lambda^{d/2})$ when $\lambda\rightarrow+\infty$. If the phase depends also on $\lambda$, say if you replace $\lambda \phi(x)$ by $\lambda\Phi(x,\lambda)$, then you keep the assumption on $a$ but you also assume that $\Phi$ is such that there exists positive constants $c_0,C_0$ such that $$ \quad d\Re \Phi=0 \text{ and } \Im\Phi=0 \Longrightarrow c_0\le \vert\det(\Phi'')\vert \le C_0. \tag{$\sharp$} $$ This is a short description of the method of complex stationary phase. Note that the integral $I$ is largest at points where the phase is realvalued and stationary (i.e. with a vanishing gradient). This is quite natural after all since as long as the phase is positive you get an exponential decay, and when it is realvalued the differential of the imaginary part is also vanishing since that imaginary part is assumed to be nonnegative. Then you must take a look at the Hessian of $\phi$, as in the method of real stationary phase for realvalued Morse function. 

