  Using the theorem of residues Chiarella and Reichel derived a series that can
be represented in terms of the complex error function (CEF). Here we show a
simple derivation of this CEF series by Fourier expansion of the exponential
function $\exp ({- {\tau ^2}/4})$. Such approach explains the existence of the
lower bound for the input parameter $y = \operatorname{Im} [z]$ restricting the
application of the CEF approximation. An algorithm resolving this problem for
accelerated computation of the CEF with sustained high accuracy is proposed.
