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Put $\eta = r+it, r>0$ and $K_{\eta}(z, w, \lambda)= \frac {\mid \lambda\mid }{2\sinh(\eta \mid \lambda \mid) } e^{-\frac{\mid\lambda\mid \coth(\eta | \lambda |) \mid z-w \mid ^{2}}{4}} e^{-i \frac{\lambda}{2} Im(z\cdot \bar{w})}$. 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,?

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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 complex-valued 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 real-valued 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 real-valued the differential of the imaginary part is also vanishing since that imaginary part is assumed to be non-negative. Then you must take a look at the Hessian of $\phi$, as in the method of real stationary phase for real-valued Morse function.

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