1970/03/01 by Walter Gautschi · 13 citations
Mathematics · Computer Science · #Mathematical functions and polynomials #Mathematical and Theoretical Analysis #Numerical Methods and Algorithms
paper · doi:10.1137/0707012
The paper is concerned with the computation of w(z) = exp ( - z2 )erfc( - iz) for complex z = x + iy in the first quadrant Q1 :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 → ∞ in the sector S: - π / 4 < arg z < 5π / 4. Specifically, the nth convergent approximates w(z) to within an error of O(z - 2n - 1 1) as z → ∞ in S. A recursive procedure is then developed which permits evaluating w(z) to a prescribed accuracy for any z ∈ Q1 . 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 / π )∫ - ∞ ^∞ exp ( - t2 )dt / (z - t) .