The first few terms of Taylor series at $a=0$ of the Gaussian function $e^{-(x-a)^2}$ when $x$ is real are:
$$e^{-x^2}+a\left(2xe^{-x^2}\right)+a^2(2x^2-1)e^{-x^2}+...$$
I am wondering if one can take the "Taylor series" at $a=0$ for the complex Gaussian function $e^{-|x-a|^2}$. Will it have the following form?
$$e^{-|x|^2}+a\left(2xe^{-|x|^2}\right)+|a|^2(2|x|^2-1)e^{-|x|^2}+...$$
I'm trying to approximate complex Gaussian around $a=0$, but am really stumbling around in the dark here, having never taken the course on complex analysis. I read about the relevant part of the wiki article on Taylor series, but am confused. Any help would be appreciated.