I am trying to find the first term in the asymptotic expansion of
$$\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{1}{s^2}e^{t(s-m\sqrt{s^2-1})} ds $$
where $0<m<1$, $c<1$, as $t$ approaches $\infty$ with $m$ fixed.
I think I am supposed to used the method of steepest descents to deform the path from a line parallel to the imaginary axis to one where $Im(s-m\sqrt{s^2-1})$ is a constant, but am unsure how to actually proceed.