Absolute Value of Error Function

Question

Let $z=a+\mathrm{i} b$ and $erf(z)=\frac{2}{\sqrt{\pi}}\int_{0}^{z}e^{-t^2}~dt$. We know \begin{align*} \overline{erf(z)}=erf(\overline{z}). \end{align*} What can we say about \begin{align*} \vert erf(z)\vert=???. \end{align*}

Answer 1

From the Wolfram fuunction site you get $$|\mathrm{erf}(a+ib)| =\sqrt{\mathrm{erf}\left(a-a\sqrt{-\frac{b^2}{a^2}}\right) \mathrm{erf}\left(a+a\sqrt{-\frac{b^2}{a^2}}\right)} $$