A rapid and highly accurate approximation for the error function of complex argument

We present efficient approximation of the error function obtained by Fourier expansion of the exponential function $\exp [{- {(t - 2 \sigma)^2}/4}]$. The error analysis reveals that it is highly accurate and can generate numbers that match up to the last decimal digits with reference values. Due to simple representation the proposed error function approximation can be utilized in a rapid algorithm.