In the method of steepest descent, we approximate to leading order an integral of the form $$I(x) = \int_C f(t) \exp(x \phi(t)) dt$$ as $x \to \infty$ by observing that if we write $t = \xi + i \eta$ and $\phi(\xi, \eta) = u(\xi, \eta) + i v(\xi, \eta)$, then if we follow the contour of steepest descent in $u$ or, equivalently, constant $v$, then we may apply Laplace's method. In the actual application of this method, however, we often must also move along contours that are not of steepest descent of $u$ or constant $v$, and ultimately we aim to show that such contours are negligible. Moreover,in practice the integrals are much more tractable if we choose to move along steepest descent contours that pass through saddle points in $u$.
Thus, my questions are:
1.) Why is it that in practice we are able to show contours that are not steepest descent end up being negligible?
2.) Why is it so important that we choose contours that pass through saddle points of $u$?