I'm trying to understand the asymptotic behavior of (analytic) error function $\mathrm{erf}(z)$ when $z\to\infty$, here the error function is defined on the whole complex plane: $$\mathrm{erf}(z)=\frac{2}{\sqrt{\pi}}\int_{\Gamma}\mathrm{e}^{-t^2}dt$$ where $\Gamma$ is the line segment connecting $0$ and $z$. This function has an essential singularity at $z=\infty$.
I'm trying to find the asymptotic expansion of $\mathrm{erf}(z)$ at $z=\infty$, this can be done by taking the expansion of $\mathrm{erf}(1/z)$ at $z=0$. I know from Wolfram Alpha that the result is $$\mathrm{erf}\Big(\frac{1}{z}\Big)=e^{-1/z^2}\left(-\frac{z}{\sqrt{\pi}}+O\left(z^2\right)\right)+(-1)^{\lfloor\mathrm{Arg}(z)/\pi+1/2\rfloor}$$
But I don't know how compute this. Any references or hints will be helpful, thanks.