I want to convolute a complex Gaussian function
$G(x)=\exp(i\alpha x^2)$
with an one-sided oscillating exponential decay looking like
$F(x) = H(x)\exp(-\lambda x -ik x)$,
where $H(x)$ is the heaviside function. After some substitutions and pushing constants back and forth, I obtain an integral, which should be proportional to my desired convolution:
$K(x) = \int_0^\infty{\exp(-\lambda z - i \beta(x) z + i \alpha z^2)\,\text{d}z}$.
And here the trouble starts: I don't know how to approach this integral. The quadratic complex term looks like I could apply Fresnel-integrals, but I am not quite sure, whether they are really useful in this case.
Does anybody have an idea in which direction to go? Or probably convincing reasons, why there is not solution to this integral?
Thanks a lot!
EDIT:
After some further searching, I found this question on the physics stack exchange. Therein they give the solution of a generalized gaussian integral as
$\int_{-\infty}^\infty {\exp (-\frac{a}{2}x^2+bx)} = \sqrt{\frac{2\pi}{a}}\exp (\frac{b^2}{2a})$,
but mention that for $\Re(a)=0$ (which is the case for my example) this solution is only applicable, if also $\Re(b)=0$ (which is not the case for my example). Ok, so I can't use this solution. But, does that probably already implicitly mean that there is no solution for me?