TITLE: erf(a+ib) error function separate into real and imaginary part
QUESTION [14 upvotes]: Is there an easy way to separate erf(a+ib) into real and imaginary part?

REPLY [3 votes]: The identities for the real and imaginary parts of the error function are:
$$
\begin{aligned}
{\mathop{\rm Re}\nolimits} \left[ {{\rm{erf}}\left( {x + iy} \right)} \right] &= \frac{{{\rm{erf}}\left( {x + iy} \right) + {\rm{erf}}\left( {x - iy} \right)}}{2}\\
{\mathop{\rm Im}\nolimits} \left[ {{\rm{erf}}\left( {x + iy} \right)} \right] &= \frac{{{\rm{erf}}\left( {x + iy} \right) - {\rm{erf}}\left( {x - iy} \right)}}{{2i}}
\end{aligned}
$$
See the paper [Abrarov and Quine, J. Math. Research, 7(2), 2015, pp. 163-174] for the rigorous proof: http://dx.doi.org/10.5539/jmr.v7n2