  In this paper we present two efficient approximations for the complex error
function $w \left( {z} \right)$ with small imaginary argument
$\operatorname{Im}{\left[ { z } \right]} < < 1$ over the range $0 \le
\operatorname{Re}{\left[ { z } \right]} \le 15$ 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.
