Efficient Multi-Accuracy Computations of Complex Functions with Complex Arguments

Abstract: We present an efficient multi-accuracy algorithm for the computations of a set of special functions of a complex argument, z=x+iy. These functions include the complex probability function w(z), and closely related functions such as the error function erf(z), complementary error function erfc(z), imaginary error function erfi(z), scaled complementary error function, erfcx(z), the plasma dispersion function Z(z), Dawson s function Daw(z), and Fresnel integrals S(z) and C(z). Computational results from the present algorithm are compared with results from competitive algorithms and widely used software packages showing superior accuracy and efficiency of the present algorithm. In particular, the present results highlight concerns about the accuracy of evaluating such special functions using commercial packages like Mathematica and free/open source packages like the MIT-C++ package.