Asymptotic expansion using method of steepest descents

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.

• You mention that your contour should be parallel to the imaginary axis, so did you intend the bounds for the integral to be $c \pm i\infty$ instead of $c\pm\infty$? – Antonio Vargas Dec 1 '15 at 9:35
• Yeah I did, that was my bad - thanks for catching that! – meanderingthroughmath Dec 2 '15 at 10:29
• The main idea is to deform your contour so that it passes through the saddle point at the relevant solution to $\frac{d}{ds} (s-m\sqrt{s^2-1}) = 0$. – Antonio Vargas Dec 2 '15 at 10:37