Accurate approximations for the complex error function with small imaginary argument
Abstract
In this paper we present two efficient approximations for the complex error function w ( z ) with small imaginary argument Im[ z ] < < 1 over the range 0 Re[ z ] 15 that is commonly considered difficult for highly accurate and rapid computation. These approximations are expressed in terms of the Dawson's integral F( x ) 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.