Rational Approximations to the Complex Error Function

Rational Approximations to the Complex Error Function

Thu, 03/12/2009
14:00 		Prof. Andre Weideman (University of Stellenbosch) Computational Mathematics and Applications Add to calendar 3WS SR
Rational Approximations to the Complex Error Function
We consider rational approximations to the Faddeeva or plasma dispersion function, defined as $ w(z) = e^{-z^{2}} \mbox{erfc} (-iz) $. With many important applications in physics, good software for computing the function reliably everywhere in the complex plane is required. In this talk we shall derive rational approximations to $ w(z) $ via quadrature, Möbius transformations, and best approximation. The various approximations are compared with regard to speed of convergence, numerical stability, and ease of generation of the coefficients of the formula. In addition, we give preference to methods for which a single expression yields uniformly high accuracy in the entire complex plane, as well as being able to reproduce exactly the asymptotic behaviour $ w(z) \sim i/(\sqrt{\pi} z), z \rightarrow \infty $ (in an appropriate sector). This is Joint work with: Stephan Gessner, Stéfan van der Walt