Expansion of the complex normal distribution around sigma = 0

Question

I would like to expand the function $$ f_\sigma(x, x') = \frac{1}{\sqrt{2\pi i \sigma^2}}e^{i\frac{(x-x')^2}{2\sigma^2}} $$ around $\sigma^2\approx 0$. I only know $$ \lim_{\sigma^2\to 0}f_\sigma(x, x') =\delta(x-x') $$ So I would expect $$ f_\sigma(x, x') \approx \delta(x-x') + O(\sigma^2) $$ How do I find the next term, namely $O(\sigma^2)$ ? Is it even possible?