Integral of third order polynomial exponential

[PHAM Duong Hung]

Hello,
I am looking for approximated or exact solution of
\begin{align}
I = \int_R \exp(cx^3-ax^2+bx)dx
\end{align}
where $a,b,c$ are complex numbers defined as:
\begin{align}
c &= \frac{1}{3}i\pi\phi'''(t) \notag\\
a &= \dfrac{1}{2\sigma^2}-i\pi \phi''(t) = re^{i\varphi}~~\text{with}~~~ r = \dfrac{1}{2\sigma^2}\sqrt{1+4\pi^2\sigma^4\phi''(t)^2} ~~\text{and}~~\varphi = arctan(-2\pi\sigma^2\phi''(t))\notag\\
b &= -i2\pi\eta
\end{align}

The fact that I computed the following :

\begin{align}
\int_{\mathbb{R}} \exp(i\alpha x^3)dx = \frac{2}{3} \frac{\alpha^{-1/3}\pi}{\Gamma(\frac{2}{3})}
\end{align}

Any help is greatly appreciated!

 

[Ssnow]

Hi, for the approximated method I suggest the stationary phase method:

https://en.wikipedia.org/wiki/Stationary_phase_approximation

The idea is '' you search the critical points of your phase $cx^3-ax^2+bx$ and after you approximate the integral near this points because they contribute in the major part for the area under the function '', this is only the idea for details you can see the link ...