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 $u$?


It sounds like OP is asking for an intuitive explanation for the method of steepest descent. The integrand is assumed to be holomorphic (away from poles and branch cuts) so we can deform the integration contour $\gamma$ as we like (picking up residues if we cross poles etc).

Heuristically, the two function $\phi$ and $f$ play first and second fiddle, respectively. The idea is now to choose a convenient contour $\gamma$, and choose points $p_1, \ldots, p_n$ along $\gamma$ so that the corresponding $n$ Gaussian approximations to $\phi$ capture the bulk of the integrand.

Now it is a general fact from calculating convergent Gaussian integrals, that they should be evaluated along a line through their saddle point in the direction of steepest descent.

Therefore $p_1, \ldots, p_n$ should be saddle points, and $\gamma$ should be in the direction of steepest descent.

(Another way of saying it is, that if $p_i$ is not a saddle point, or if it is not in the direction of steepest descent, then the choice is not optimal: there will be a more dominant contribution from an infinitesimally nearby point/contour.)

Conversely with this prescription, for large $x$, the integrand away from the saddle points $p_1, \ldots, p_n$ is suppressed, so that the Gaussian approximations around $p_1, \ldots, p_n$ carry the ballpark of the integrand.


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