Happy New Year to you all.
Let $\tag 1 J(N)=\int_a^b e^{Nf(x)}dx$ where $N\in\mathbb R$ and $N>>1$ and $f(x)$ has a global maximum at $x=x_0$ with Taylor expansion $f(x) \approx f(x_0)-|f''(x_0)|\frac{(x-x_0)^2}{2}.$ Then by the saddle point approximation, we get $ \tag 2 J(N)\approx \frac{e^{Nf(x_0)}}{\sqrt{N}}\sqrt{\frac{2\pi}{|f''(x_0)|}}\left(1+O\left(\frac{1}{N}\right)\right).$
Let $\tag 3 Ai(x)= \int_\gamma \exp{\left(tx-\frac{t^3}{3}\right)}dt$ where $\gamma$ is a contour in $\mathbb{C}$ chosen such that the integral converges and the integrand vanishes as $t\to\infty$.
First question: For $x>0$, I want to find an approximation to (3). So, by letting $t=u \sqrt{x}$ , I get $ Ai(x) = \sqrt{x} \int_{\mathbb{R}_+}\exp{\left(\left(u -\frac{u^3}{3}\right)x^{3/2}\right)}$ Doing so allows me to identify the following:$ N= x^{3/2}~~~f(u)=u-u^3/3~~~$ from which I get $ Ai(x>0) = \sqrt{2\pi}e^{{2/3}z^{3/2}}x^{-1/4}\left(1+O\left(z^{-3/2}\right)\right).$ However, the solution is given as $ Ai(x>0) = \sqrt{2\pi}e^{{-2/3}z^{3/2}}x^{-1/4}\left(1+\left(x^{-3/2}\right)\right).$ Obviously, the my trouble stems from the fact that using (2), I keep getting 2/3 and not -2/3. I'd be very happy if someone could point me in the right direction.
Second question: For $x<0$, can I get help in using the stationary phase approximation to approximate (2)?
Please, if what I've done above is complete hokum, let me know as well. Thanks.