Efficient application of the Chiarella and Reichel series approximation of the complex error function

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.