I am trying to analytically Fourier transform a set of functions that have the form
$f(k) e^{-\rho~\psi(k)}$
where the general $f(k)$ is some linear combination of trig functions without poles, $\psi(k) = - a \sin(k)^{2} + i~c~ (\cos(k)^{2}+\gamma^{2}\sin(k)^{2})^{1/2}$ and $\rho \gg 1$. The parameters $a \gt 0,c$ are all real and order one, while $ 0 \lt \gamma \lt 1 $ and real.
The issue is that I want to find the fourier transform/ fourier component that also has this large scale $\rho$ in it, i.e the factor in the integrand for the FT of $e^{i k x} = e^{\rho~i k b}$. The form of the integral then becomes
$\int^{\pi/2}_{-\pi/2} f(k) ~e^{\rho~g(k)} dk$,
where $g(k) = - a \sin(k)^{2} + i~c~ (\cos(k)^{2}+\gamma^{2}\sin(k)^{2})^{1/2} + i b k$
My first idea was to use integral asymtopics, method of steepest descent since $\rho \gg 1$ or approximate about the saddle points but I get stuck right away at finding the saddle points. The problem I find is dealing with the root in the third term when finding the saddle points or the imaginary part (to find the contours of constant phase). For example, differentiating gives
$g '(k) = i b -a \sin(2k) + \dfrac{i c (\gamma^{2}-1)\sin(2k)}{2(\cos(k)^{2}+\gamma^{2}\sin(k)^{2})^{1/2}} = 0$
which when squared leads to a 6th order polynomial, once all the trig functions have been written in terms of $\tan(k)$. Explicitly equating real and imaginary parts also leads to difficult equations.
Also, the function is well approximated by replacing $-a \sin(k)^{2} \to -a k^{2}$ which might help but then you have to deal with a polynomial term with the trig terms.
Does this look possible to evaluate analytically in this way? Or are there other techniques that can be used? Mathematica gives the solution in terms of its root functions, so doesn't seem that helpful.