- #1
PHAM Duong Hung
- 1
- 0
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!
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!