I am being asked to show that the saddle point of the following integral is at $t_0 = \frac{\pi i}{2}$ and that the steepest descent path through this point can be approximated in the neighbourhood of $t = \frac{\pi i}{2}$ by the straight line $t = x + i(\frac{\pi}{2} + x)$. The integral is: $$H_\nu^{(1)}(z) = \frac{1}{\pi i} \int_{-\infty}^{\infty + \pi i} e^{z \sinh t - \nu t} \, dt$$
First of all, where does the $x$ come from? I suspect there is a substitution, because when you equate the derivative of the exponent to $0$, which is what afaik you have to do to get the saddle point, you get an answer in terms of $v$. So what substitution do I need to make, and how do I proceed once I have the required saddle point?