I'm trying to evaluate(approximate) the following integral
$$ F(x,t;q) = \int_{-\infty}^{\infty}\frac{q}{q+2ik} e^{i(kx +8k^3 t)}\; dk $$ It's similar to the Airy function but I can't get rid of the $\frac{1}{k}$ in front of the exponential.
$q>0 $ is a parameter that I can vary.
$x \in (-\infty,\infty)$ is my spatial variable
$t > 0$ is my time dependence
Any ideas?
An approximate solution in the large or small $q$ limit would also be fine.
Since $\frac{\lambda}{\lambda-it}$ is the CF of the exponential distribution while $e^{ik^3}$ is the Fourier transform of the Airy function $Ai(-x)$, your integral is a convolution integral that can be evaluated through the exponential integral $E_{n}(x)$ for $n=-\frac{1}{2},-\frac{2}{3}$ or just through the incomplete gamma function $\Gamma\left(\frac{4}{3},x\right),\Gamma\left(\frac{5}{3},x\right).$