Airy transform is defined as follows:
$$ f_a(x)=\int_{-\infty}^{\infty}f(t)\mathrm{Ai}(t-x)dt $$
According to the book by Olivier Vallee and Manuel Soares it has the translation property: $f_a(x+c)$ is the Airy transform of $f(x+c)$. I can see that this is valid by just change of variables. The integration limits won't change. However if the translation is complex valued then the integration limits would be shifted in the complex domain:
$$\int_{-\infty}^{\infty}f(t+ic)\mathrm{Ai}(t-x)dt= \int_{-\infty+ic}^{\infty+ic}f(t')\mathrm{Ai}(t'-(x+ic))dt' $$
So my question is that if the translation property is valid for the complex shift?
I am particularly interested in the case where $f$ is a Gaussian function. I have numerically checked and it seems that it is valid.