Integral of third order polynomial exponential

  • #1
I am looking for approximated or exact solution of
I = \int_R \exp(cx^3-ax^2+bx)dx
where $a,b,c$ are complex numbers defined as:
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

The fact that I computed the following :

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

Any help is greatly appreciated!

Answers and Replies

  • #2
Gold Member
Hi, for the approximated method I suggest the stationary phase method:

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 ...