I am a physicist trying to calculate the lineshape $f_2(\nu)$ for the oscillations given in a new dark matter model; the entire problem at the end can be reduced to calculating a convolution,
$f_2(\nu) = \int_{-\infty}^{+\infty} f_1(\phi)f_1(\nu + \phi)d\phi$,
where
$f_1(\nu) = \alpha e^{-\beta \nu}\sinh(\gamma \sqrt{\nu-\delta})$.
$\alpha, \beta, \gamma$ and $ \delta $ are real, positive constants.
This yields a very difficult integral (atleast for me), that I would be very interested in solving:
$f_2(\nu) = \alpha^2 \int^{\infty}_{-\infty} e^{-\beta( 2\phi + \nu)} \sinh(\gamma \sqrt{\phi-\delta}) \sinh(\gamma\sqrt{\phi+\nu-\delta}) d\phi$
I tried expanding the hyperbolic sines using the identity $\sinh(a)\sinh(b) = \frac{1}{4}(e^a-e^{-a})(e^b-e^{-b})$ in the hopes that having everything expressed as an exponential would hint at something; it didn't.