Accurate approximations for the complex error function with small imaginary argument

In this paper we present two efficient approximations for the complex error function $w\left(z\right)$ with small imaginary argument $\mathrm{Im}\left[z\right]<<1$ over the range $0\le \mathrm{Re}\left[z\right]\le 15$ that is commonly considered difficult for highly accurate and rapid computation. These approximations are expressed in terms of the Dawson’s integral $F\left(x\right)$ of real argument $x$ that enables their efficient implementation in a rapid algorithm. The error analysis we performed using the random input numbers $x$ and $y$ reveals that in the real and imaginary parts the average accuracy of the first approximation exceeds ${10}^{-9}$ and ${10}^{-14}$, while the average accuracy of the second approximation exceeds ${10}^{-13}$ and ${10}^{-14}$, respectively. The first approximation is slightly faster in computation. However, the second approximation provides excellent high-accuracy coverage over the required domain.