# Higher order corrections to saddle point approximation

I'd like to ask for hints how to obtain higher order corrections to approximations obtained by the saddle point method. References will be also welcome. Unfortunately what comes up when googling is mostly just usual leading order approximation.

Let me show my idea how to do it. Consider an integral $I(t)= \int_{\mathbb R} e^{tx^2 - x^4} dx$. I am interested in the limit $t \to \infty$ through complex values. My idea is to expand quartic in the exponential around its stationary point $x_0$ as $a+b(x-x_0)^2-x^4$ and replace $e^{-x^4}$ by $1-x^4$. The result is that I get correct leading order behaviour, but wrong next to leading order corection for some values of argument of $t$. I know that the next to leading order term is wrong because I know the exact form of this integral in terms of Bessel functions and asymptotics of these are known.

A very instructive reference for this is section 4.7 of Miller's Applied Asymptotic Analysis which performs the analogous analysis to determine the asymptotic behavior of the Airy function. Carrying out the full analysis of your integral would be too much for an answer here, so I'll give an outline and the appropriate section references from the book.

First write $t = re^{i\theta}$ with $r \geq 0$ and substitute $x = \sqrt{r} y$ to get

$$I(t) = \sqrt{r}\int_{\mathbb R} \exp\!\left\{ r^2 \left(e^{i\theta} y^2-y^4\right)\right\}dy.$$

The exponent function $\varphi_\theta(y) = e^{i\theta}y^2 - y^4$ has three saddle points $y = y^*$, one at $y^*=0$ and one at either solution of $(y^*)^2 = e^{i\theta}/2$.

Depending on the value of theta, one (or more) of these saddle points $y^*$ of $\varphi_\theta$ will dominate the others and determine the asymptotics. This is known as the Stokes phenomenon. The process of determining which saddle point dominates is described in section 4.7 of the book.

Once the appropriate saddle points are determined it remains to apply the method of steepest descent as described in sections 4.2 through 4.4. The basic idea is that after the appropriate contour has been chosen you apply the Laplace method as described in section 3.4.

This process yields a complete asymptotic expansion which is valid as $r \to \infty$ for a given fixed $\theta$.