Efficient Computation of the Complex Error Function

Abstract

The paper is concerned with the computation of $w(z) = \exp ( - z^2 ){\operatorname{erfc}}( - iz)$ for complex $z = x + iy$ in the first quadrant $Q_1 :x \geqq 0,y \geqq 0$. Using Stieltjes– theory of continued fractions it is first observed that the Laplace continued fraction for $w(z)$, although divergent on the real line, represents $w(z)$ asymptotically for $z \to \infty $ in the sector $ S:{{ - \pi} / 4} < \arg z < {{5\pi } / 4}$. Specifically, the nth convergent approximates $w(z)$ to within an error of $O(z^{ - 2n - 1} 1)$ as $z \to \infty $ in S. A recursive procedure is then developed which permits evaluating $w(z)$ to a prescribed accuracy for any $z \in Q_1 $. The procedure has the property that as $| z |$ becomes sufficiently large, it automatically reduces to the evaluation of the Laplace continued fraction, or, equivalently, to Gauss–Hermite quadrature of $({i / \pi })\int_{ - \infty }^\infty {\exp ( - t^2 ){{dt} / {(z - t)}}} $.