Background
I have recently come very close to an analytical solution to a particular PDE related to my research. In particular, solving a very specific case of the Navier-Stokes equations in a torus with a variable viscosity field. Results for the case with constant viscosity actually exist and I am confident in my derivation of the equations with variable viscosity, as I have applied the same method to rederiving the equation with constant viscosity, as well as other known equations with variable viscosity (i.e. cartesian and cylindrical coordinates). This particular equation comes from transforming the original into the toroidal coordinates $ \tau $ and $ \sigma $.
The equation ends up being just nearly separable, with one pesky term left that I have actually proven that there exists no additional transformation/change of variables that will make it separable that won't also heavily restrict what the viscosity field can be. My idea now is to be able to take the Fourier Transform of both sides in the variable $ \sigma $, get an ODE for $ \tau $ and then transform back. The reason Fourier transform seems so tempting is because the only $ \sigma $ derivative in the equation comes as the second derivative with no coefficient attached. Namely, the portion I'm talking about and the problem term together are:
$$ U_{\sigma \sigma}(\tau,\sigma) $$ and $$ \frac{\cosh(\tau)\cos(\sigma)-1}{\cosh(\tau)-\cos(\sigma)}U(\tau,\sigma) $$
Where I am using the subscript notation here as the derivative. Since all other terms in the equation are only $ U(\tau,\sigma) $ and its derivatives in $ \tau $ with coefficients in terms of $\tau$ only along with the fact that the Fourier Transform of the second derivative in $\sigma$ is just $-\omega^2 \hat{U}$, I really want to see if taking the Fourier transform in this extra term is a possible route.
The Fourier Transform
In particular, I am interested in calculating: $$ \int_{-\infty}^{\infty} \frac{\cosh(\tau)\cos(\sigma)-1}{\cosh(\tau)-\cos(\sigma)}U(\tau,\sigma) e^{-i\omega\sigma} d\sigma$$ Some preliminary observations are that $ U(\tau,\sigma)$ is safely not singular on the $\sigma$ real line, as well as being $2\pi$ periodic in $\sigma$. In addition, the denominator $\cosh(\tau)-\cos(\sigma)$ is only singular when $\cosh(\tau) = \cos(\sigma)$. Since $\tau$ is real, this can only occur when $\tau = 0$ such that $\cosh(\tau) = 1$, giving singularities at integer multiples of $2\pi$. However, this is not a problem as $\tau = 0$ corresponds to a torus of infinite diameter, which the bounds of my domain of interest do not include. So with this, am I safe in saying that this Fourier Transform is well defined?
I have actually tried evaluating this integral using the Residue Theorem, but it quickly led down a road to nowhere through an infinite number of residues from a contour traversed an infinite number of times. To be clear I attempted to use the transformation $z=e^{i\sigma}$ and traverse the unit circle and infinite number of times from $0$ to $2\pi$. Is there another way to do this choosing the contour to enclose an entire upper (or lower) half plane? Also, I assumed that the $\sigma$ part of $U$ was expandable in complex Fourier series, so I could calculate the residues associated with that. However, the coefficients of this series I am not sure would be able to be calculated...
Any insight on the possibility of this approach, errors in my logic, or another approach are greatly appreciated. Thanks!